Question: Find m \\angle CAM.

Answer: 28

1. Let the right angle of the upper triangle be ∠ABM, and the right angle of the lower triangle be ∠TDM. According to the definition of a right triangle, ∠ABM = ∠TDM = 90°, so both triangles ABM and TMD are right triangles. According to the congruence criterion for right triangles (hypotenuse and one leg), we have MT = MA (both are radii of the circle) and MB = MD, so triangles TDM and MBA are congruent.

2. According to the definition of congruent triangles, ∠MTD = ∠CAM = 28°.

圆; 垂线; 圆内接四边形; 圆周角; 对顶角

Two right triangles are congruent if their hypotenuses and one corresponding leg are equal.

In <note>right triangle TDM</note> and <note>right triangle MBA</note>, <note>angle ABM and angle TDM are both right angles (90 degrees)</note>. If <note>side MA is equal to side MT (hypotenuse)</note>, and <note>side MD is equal to side MB (leg)</note>, then according to the <note>Right Triangle Congruence (Hypotenuse, One Leg)</note> principle, these two <note>right triangles TDM and MBA are congruent</note>.

Definition of Congruent Triangles

Two triangles are congruent if and only if their corresponding sides are equal and their corresponding angles are equal.

<note>Triangle TDM and triangle MBA are congruent triangles</note>, the corresponding sides and corresponding angles of triangle TDM are equal to those of triangle MBA, namely: <note>side MT = side MA</note>, <note>side MD = side MB</note>, <note>side TD = side AB</note>, and the corresponding angles are also equal: <note>angle MTD = angle MAB</note>, <note>angle MDT = angle MBA</note>, <note>angle TMD = angle AMB</note>.

Definition of Radius

<note>In circle M, point M is the center of the circle</note>, point A is any point on the circle, <note>line segment MA is the line segment from the center to any point on the circle</note>, therefore <note>line segment MA is the radius of the circle</note>. <note>In circle M, point M is the center of the circle</note>, point T is any point on the circle, <note>line segment MT is the line segment from the center to any point on the circle</note>, therefore <note>line segment MT is the radius of the circle</note>.

Definition of Right Triangle

A triangle is classified as a right triangle if it has one interior angle measuring exactly 90 degrees.

<note>In triangle ABM, angle ABM is a right angle (90 degrees)</note>, therefore triangle ABM is a right triangle. <note>Side AB and side MB are the legs</note>, <note>side MA is the hypotenuse</note>. <note>In triangle TMD, angle TDM is a right angle (90 degrees)</note>, therefore triangle TDM is a right triangle. <note>Side TD and side MD are the legs</note>, <note>side MT is the hypotenuse</note>.


Question: A B \\perp D C and G H \\perp F E. If \\triangle A C D \\sim \\triangle G E F, find A B.

Answer: 2.2

1. Given that AB is perpendicular to DC and GH is perpendicular to FE, according to the definition of perpendicular lines, ∠ABC=∠GHE=90°.

2. According to the problem, it is known that △ACD ∽ △GEF, that is, △ACD and △GEF are similar triangles, so according to the definition of similar triangles: the corresponding angles of △ACD and △GEF are equal, and the corresponding sides are proportional.

3. Because ∠C=∠E (corresponding angles in △ACD ∽ △GEF are equal), ∠ABC=∠GHE, therefore △ABC and △GHE are similar (according to the AA criterion for similar triangles), so AC/GE=AB/GH (corresponding sides are proportional).

4. Substitute the known values AC=4.4, GE=6.3, GH=3.15 into the proportional relationship, and get 4.4 / 6.3 = AB / 3.15.

5. Solve the proportion equation, cross-multiplying to get AB = 4.4 × 3.15 / 6.3.

6. Calculate to get AB = 2.2.

7. The final answer is 2.2.

垂线; 直角三角形; 线段

Definition of Similar Triangles

Two triangles are similar if and only if their corresponding angles are equal and their corresponding sides are proportional.

<note>Triangle ACD and triangle GEF are similar triangles</note>. According to the definition of similar triangles: <note>∠CAD = ∠EGF</note>, <note>∠ACD = ∠GEF</note>, ∠ADC = ∠GFE; <note>AC/GE = AD/GF</note> = CD/EF. <note>Triangle ABC and triangle GHE are similar triangles</note>. According to the definition of similar triangles: ∠CAB = ∠EGH, ∠ACB = ∠GEH, ∠ABC = ∠GHE; <note>AC/GE = AB/GH = CB/EH</note>.

Definition of Perpendicular Lines

Two lines are said to be perpendicular if and only if the angle formed by their intersection is 90 degrees.

<note>In triangle ADC, the angle ∠ABC formed by the intersection of line AB and line CD is 90 degrees</note>, therefore, according to the definition of perpendicular lines, line AB and line CD are perpendicular to each other. <note>In triangle GEF, the angle ∠GHE formed by the intersection of line GH and line FE is 90 degrees</note>, therefore, according to the definition of perpendicular lines, line GH and line FE are perpendicular to each other.

Similarity Theorem for Triangles (AA)

Two triangles are similar if two of their corresponding angles are congruent.

In <note>triangle ABC</note> and <note>triangle GHE</note>, <note>angle ABC is equal to angle GHE</note>, and <note>angle C is equal to angle E</note>, so <note>triangle ABC is similar to triangle GHE</note>.


Question: Find x.

Answer: 10 \sqrt { 3 }

1. Given that the legs of a right triangle are 10 and x respectively, and the hypotenuse is 20, according to the Pythagorean theorem (in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs), we can obtain the following equation: 10^2 + x^2 = 20^2.

2. Substitute the known values into the equation of the Pythagorean theorem to get: 100 + x^2 = 400.

3. Solve the above equation to get: x^2 = 300.

4. Take the square root of both sides of the equation to get: x = sqrt(300) or x = -sqrt(300).

5. Since x represents the length of a side, it cannot be negative, so we take the positive value of x: x = sqrt(300) = 10√3.

6. Through the above reasoning, the final answer is x = 10√3.

直角三角形; 余弦; 中点

Definition of Right Triangle

<note>This triangle is a right triangle</note>, in which <note>one interior angle is 90 degrees</note>. The side lengths are <note>the two sides with lengths 10 and x are the legs</note>, and the side length of <note>20 is the hypotenuse</note>.

Hypotenuse of a Right Triangle

In a right triangle, the hypotenuse is the side opposite the right angle, and it is also the longest side among the three sides of the triangle.

<note>Hypotenuse is AB</note>, with a length of 20.

Right Triangle Legs

In a right triangle, the two sides other than the hypotenuse are called the legs.

<note>The right triangle legs are AC and BC</note>, where <note>AC=10</note>, <note>BC=x</note>.

Pythagorean Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs.

In the figure of this problem, in the <note>right triangle</note>, the <note>legs are 10 and x respectively</note>, and the <note>hypotenuse is 20</note>, so according to the <note>Pythagorean Theorem</note>, ##10^2 + x^2 = 20^2##.


Question: X Y and X Z are the medians of 	riangle R S T. Find S T.

Answer: 14

1. Given that XY and XZ are the medians of △RST, the points X, Y, and Z are the midpoints of the three sides of triangle RST. According to the Midsegment Theorem, the segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long. Therefore, the segment connecting the midpoints of segments RS and RT, segment XY, is parallel to segment ST and XY is half of ST. That is, XY∥ST, XY = 1/2 × ST.

####

2. Additionally, given in the figure that XY = 7, which is the midsegment XY, therefore, according to the Midsegment Theorem, ST = 2 × XY = 2 × 7.

3. Finally, through the above reasoning, we conclude that ST = 14.

中点; 平行线; 普通三角形

Triangle Midline Theorem

In a triangle, if a line segment connects the midpoints of two sides, then this line segment is parallel to the third side and is equal to half the length of the third side.

In the diagram of this problem, in triangle RST, <note>point Y is the midpoint of side RT</note>, <note>point X is the midpoint of side RS</note>, <note>segment XY connects these two midpoints</note>. According to the Triangle Midline Theorem, <note>segment XY is parallel to the third side ST and equals half of the third side ST</note>, that is, <note>XY || ST</note>, and <note>XY = 1/2 * ST</note>.

Midpoint of a Line Segment

A midpoint of a line segment is the point that divides the line segment into two equal parts.

In the figure of this problem, <note>the midpoint of line segment SR is point X</note>. According to the definition of the midpoint of a line segment, point X <note>divides line segment SR into two equal parts</note>, that is, <note>the lengths of line segment SX and line segment RX are equal</note>. That is, <note>SX = RX</note>. Similarly, <note>the midpoint of line segment RT is point Y</note>. According to the definition of the midpoint of a line segment, point Y <note>divides line segment RT into two equal parts</note>, that is, <note>the lengths of line segment RY and line segment TY are equal</note>. That is, <note>RY = TY</note>.


Question:
Which of the following statements is correct?

A. AB < BC < AC
B. BC < AB < AC
C. AB < AC < BC
D. AC < BC < AB
E. BC < AC < AB

Answer: BC < AB < AC

1. In triangle ABC, it is known that ∠B is 61°, ∠A is 59°. Since the sum of the interior angles of a triangle is 180°, we can calculate the degree of ∠C according to the triangle interior angle sum theorem, i.e., ∠C = 180° - ∠A - ∠B = 180° - 59° - 61° = 60°.

2. In a triangle, the length of a side corresponds to the size of the opposite angle: that is, the side opposite the larger angle is longer. This is known from the sine theorem, which states that in a triangle, if one angle is larger than another angle, then the side opposite the larger angle is correspondingly longer.

3. Comparing the size relationships between the angles of triangle ABC, it is known that ∠C = 60°, ∠B = 61°, and ∠A = 59°, so ∠B > ∠C > ∠A.

4. According to the sine theorem, the side opposite ∠B, AC, is the longest, the side opposite ∠C, AB, is the second longest, and the side opposite ∠A, BC, is the shortest. Therefore, the size relationship of the sides is BC < AB < AC.

5. Comparing the options, the correct order is option B: BC < AB < AC.

6. After the above reasoning, the final answer is BC < AB < AC.

Triangle Angle Sum Theorem

The sum of the interior angles of any triangle is 180°.

In the figure of this problem, in triangle ABC, <note>angle A, angle B, and angle C are the three interior angles of triangle ABC</note>. According to the Triangle Angle Sum Theorem, <note>angle A + angle B + angle C = 180°</note>.

Sine Theorem

In any triangle, the ratio of the length of each side to the sine of its opposite angle is constant and equal to the diameter of the circumcircle, i.e., \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2r = D \) (where \( r \) is the radius of the circumcircle, and \( D \) is its diameter).

In <note>any triangle ABC</note>, the sides <note>AB</note>, <note>AC</note>, and <note>BC</note> correspond to <note>angle C</note>, <note>angle B</note>, and <note>angle A</note> respectively. According to the <note>Sine Theorem</note>, the ratio of the length of each side to the sine of its opposite angle is equal and is equal to the diameter of the circumscribed circle, that is: <note>AB/sin(angle C)=AC/sin(angle B)=BC/sin(angle A)</note> = 2r = <note>D</note> (where <note>r is the radius of the circumscribed circle</note>, <note>D is the diameter</note>).


Question: In triangle PQR, ZQ = 3a - 11, ZP = a + 5, PY = 2c - 1, YR = 4c - 11, m\angle PRZ = 4b - 17, m\angle ZRQ = 3b - 4, m\angle QYR = 7b + 6, and m\angle PXR = 2a + 10. If RZ is the angle bisector, find m\angle PRZ.

Answer: 35

1. Given RZ is the angle bisector of ∠QRP, according to the definition of the angle bisector, ∠PRZ = ∠ZRQ. That is: 4b - 17 = 3b - 4.

2. Solve the equation 4b - 17 = 3b - 4, rearranging terms to obtain b = 13.

3. Substitute b = 13 into the expression for ∠PRZ, 4b - 17, to get ∠PRZ = 4×13 - 17 = 52 - 17 = 35.

4. Through the above reasoning, the final answer is ∠PRZ = 35°.

普通三角形; 对顶角; 邻补角; 线段

Definition of Angle Bisector

An angle bisector is a line that originates from the vertex of an angle and divides the angle into two congruent angles.

In the diagram of this problem, <note>ZR is the angle bisector of angle QRP</note>, <note>the vertex of angle QRP is point R</note>, from <note>point R a line RZ is drawn</note>, <note>this line divides angle QRP into two equal angles</note>, namely <note>angle PRZ and angle ZRQ are equal</note>.


Question: Find the perimeter of the triangle. Round to two decimal places.

Answer: 8.73

1. Observe the geometric figure in the diagram and identify the given conditions: the triangle is an isosceles triangle, the base length is 3.5 ft, let the three vertices of the triangle be A, B, C, then ∠ACB equals 48°.

2. According to the definition of an isosceles triangle, the lengths of the two legs are equal, i.e., AB = AC. Based on the properties of an isosceles triangle, the angles opposite the equal sides are equal, thus ∠ABC = ∠ACB = 48°.

3. Since the sum of the interior angles of a triangle is 180°, applying the triangle angle sum theorem, we get: ∠BAC = 180° - ∠ABC - ∠ACB = 180° - 48° - 48° = 84°.

4. Using the sine rule, i.e., a/sin(A) = b/sin(B) = c/sin(C), where a is the length of a side of the triangle, and A is the angle opposite that side, we calculate the unknown equal leg length (in this problem corresponding to AB and AC). We assume AB = AC = x.

5. Since the base (BC) is known, we can use the sine rule formula: x/sin(48°) = 3.5/sin(84°).

6. Calculate x: x = (3.5 * sin(48°)) / sin(84°) = 3.5 * 0.7431 / 0.9945 ≈ 2.617 ft.

7. Given the perimeter of the triangle is the sum of its sides, the perimeter of triangle ABC is: AB + AC + BC = x + x + 3.5 = 2 * 2.617 + 3.5 ≈ 8.734 ft.

8. Round the answer to the nearest hundredth, thus we get the final answer as 8.73 ft.

等腰三角形; 普通三角形

Definition of Isosceles Triangle

A triangle is an isosceles triangle if and only if it has at least two congruent (≅) sides.

In the figure of this problem, <note>triangle ABC</note>, <note>sides AB and AC are equal</note>, therefore <note>triangle ABC is an isosceles triangle</note>.

Properties of Isosceles Triangle

In an isosceles triangle, the angles opposite the equal sides are equal, and the sides opposite the equal angles are congruent.

<note>In isosceles triangle ABC, sides AB and AC are equal</note>. Therefore, <note>according to the properties of isosceles triangles, the angles opposite the equal sides are equal, i.e., ∠ABC = ∠ACB = 48°</note>.

Triangle Angle Sum Theorem

In the figure of this problem, in triangle ABC, <note>angle BAC, angle ABC, and angle ACB are the three interior angles of triangle ABC</note>, according to the Triangle Angle Sum Theorem, <note>angle BAC + angle ABC + angle ACB = 180°</note>.

Sine Theorem

<note>The two legs of an isosceles triangle are AB and AC respectively, the base is BC, the base angles are ∠ABC and ∠ACB, and the apex angle is ∠BAC</note>. According to <note>Sine Theorem, the ratio of the lengths of the sides to the sine values of their opposite angles is equal</note>, namely: <note>AB/sin(∠ACB) = BC/sin(∠BAC) = AC/sin(∠ABC)</note>. Substituting values: <note>x/sin(48°) = 3.5/sin(84°)</note>, thus we can obtain <note>x ≈ 2.617 feet</note>.

Formula for the Perimeter of a Triangle

The perimeter of a triangle is the sum of the lengths of its three sides. Therefore, if the three sides of the triangle are denoted as \( a \), \( b \), and \( c \), then the perimeter \( P \) is given by \( P = a + b + c \).

The original text: <note>The three sides of triangle ABC are AB, AC, and BC respectively</note>, according to the formula for the perimeter of a triangle, the perimeter of the triangle is equal to the sum of the lengths of its three sides, i.e., perimeter <note>P=AB + AC + BC</note>.


Question: Each pair of polygons is similar. Solve for x.

Answer: 8

1. Given that the two triangles are similar, according to the definition of similar triangles, we can obtain the ratio of the corresponding sides: 2x/12=20/15.

2. Solving the equation, we get: 15x = 120.

3. Finally, x = 120 / 15 = 8.

4. Therefore, the final answer obtained is x=8.

Definition of Similar Triangles

Original text: <note>In the figure of this problem, triangle ABC and triangle DEF are similar triangles.</note> According to the definition of similar triangles: <note>∠A = ∠D, ∠B = ∠E, ∠C = ∠F; AB/DE = BC/EF = CA/FD.</note> The specific corresponding relationships are:
The side lengths of the first triangle ABC are 12, 2x, 20, and the side lengths of the second triangle DEF are 12, y+4, 15.
The corresponding sides are proportional, so the proportional relationship is:
<note>20/15 = 12/(y + 4) = 2x/12 = 4/3</note>####


Question: Using the parallelogram J K L M, find m \angle J K L.

Answer: 80

1.##According to the theorem of supplementary adjacent angles of a parallelogram, we get: ∠JKL+∠KLM=180°##

2.##According to the given conditions, ∠KLM=∠JLK+∠JLM=70°+30°=100°.##

3.##Given ∠KLM=30°+70°=100°, substituting it into the equation, we get ∠JKL +100°=180°.##

4.##Calculating, we get ∠JKL=180°-100°=80°.##

####

##5##. Through the above reasoning, the final answer is 80°.##

平行四边形; 对顶角; 三角形的外角

Definition of Parallelogram

A quadrilateral is a parallelogram if and only if it has two pairs of opposite sides that are both parallel (∥) and congruent (≅).

<note>Quadrilateral JKLM is a parallelogram</note>. According to the definition of a parallelogram, <note>side JK is parallel and equal to side LM</note>, <note>side JM is parallel and equal to side KL</note>.

In a parallelogram, each pair of adjacent interior angles are supplementary, meaning their sum is 180°.

In <note>parallelogram JKLM</note>, <note>angle KLM</note> and <note>angle JKL</note> are adjacent interior angles. According to the <note>Adjacent Angles Supplementary Theorem of Parallelogram</note>, <note>angle KLM + angle JKL = 180°</note>.


Question: Find P T.

Answer: \frac { 20 } { 3 }

1. From the figure, ∠PRQ = 90°, according to the definition of a right triangle, △PRQ is a right triangle.

2. In ΔPRQ, according to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the two legs, we get: PQ? = PR? + RQ?.

3. Substitute the given data, PQ = 25, RQ = 10 + 5 = 15, substitute into the equation: 25? = PR? + 15?.

4. Calculate to get: 625 = PR? + 225, thus we get PR? = 400.

5. By calculating the square root, we get PR = √400 = 20.

6. Since TS is parallel to PQ and RS = 10, according to the theorem of proportional segments in parallel lines, TR : PR = RS : RQ = 10 : 15.

7. From this proportion, we know TR = (10/15) * 20.

9. From the figure, we know PT = PR - TR.

10. Substitute the calculated values: PT = 20 - 40/3.

直角三角形; 等边三角形; 线段

Definition of Right Triangle

A triangle is classified as a right triangle if it has one interior angle measuring exactly 90 degrees.

<note>In triangle ΔPRQ, angle ∠PRQ is a right angle (90 degrees)</note>, therefore <note>triangle ΔPRQ is a right triangle</note>. <note>Side PR and side RQ are the legs</note>, <note>side PQ is the hypotenuse</note>.

Proportional Segments Theorem

If a line is parallel to one side of a triangle and intersects the other two sides or their extensions, then it divides the segments it intersects proportionally in relation to the corresponding segments of the original triangle.

In the figure of this problem, in triangle PRQ, <note>line TS is parallel to side PQ</note>, and <note>it intersects the other two sides PQ and RQ (or their extensions) at points T and S</note>. Then, according to the Proportional Segments Theorem, we have: <note>TR : PR = RS : RQ</note>, that is, <note>the intercepted segments are proportional to the corresponding segments of the original triangle</note>.

Pythagorean Theorem

<note>In the right triangle PRQ, ∠PRQ is a right angle (90 degrees)</note>, the sides <note>PR and RQ are the legs, and the side PQ is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>PQ² = PR² + RQ²</note>, that is, <note>25² = PR² + 15²</note>.


Question: In the figure, m \angle 4 = 101. Find the measure of \angle 6.

Answer: 101

1. Given ∠ 4 = 101°, and the two lines indicated by the red arrows are parallel to each other, according to the definition of alternate interior angles and the parallel postulate 2, the angles ∠ 4 and ∠ 6 are alternate interior angles under the same transversal line, therefore ∠ 6 = ∠ 4.

2. Substituting the known condition ∠ 4 = 101°, we obtain ∠ 6 = 101°.

3. Through the above reasoning, the final answer is 101°.

对顶角; 内错角; 平行线

Definition of Alternate Interior Angles

When a straight line (referred to as a transversal) intersects two parallel lines, the angles that are located between the two parallel lines but on opposite sides of the transversal are called alternate interior angles.

Two parallel lines are intersected by a transversal, where angle 4 and angle 6 are located between the two parallel lines, and on opposite sides of the transversal, therefore angle 4 and angle 6 are alternate interior angles. Alternate interior angles are equal, that is, angle 4 is equal to angle 6.

Parallel Postulate 2 of Parallel Lines

If two parallel lines are cut by a transversal, then the corresponding angles are equal, the alternate interior angles are equal, and the consecutive interior angles on the same side of the transversal are supplementary.

<note>Two parallel lines are intersected by a third line</note>, forming the following geometric relationships: 1. Corresponding angles: <note>Angle 4 and Angle 8 are equal</note>. 2. Alternate interior angles: <note>Angle 4 and Angle 6 are equal</note>. 3. Consecutive interior angles: <note>Angle 4 and Angle 5 are supplementary, that is, Angle 4 + Angle 5 = 180 degrees</note>. These relationships illustrate that when two parallel lines are intersected by a third line, corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary.


Question: As shown in the figure, PB is tangent to ⊙O at point B, PO intersects ⊙O at point E, the extension of PO intersects ⊙O at point A, connect AB, the radius OD of ⊙O is perpendicular to AB at point C, BP=6, ∠P=30°, then the length of CD is ()

Answer: √{3}

1. Given that PB is tangent to ⊙O at point B, ##connect BO##, according to ##the property of the tangent to a circle##, we get ∠OBP = 90°.

2. In △POB, given BP = 6, ##∠OPB = 30°, according to the triangle angle sum theorem, we get ∠POB = 180° - ∠OPB - ∠OBP = 180° - 30° - 90° = 60°##.

3. Using the tangent function formula OD = OB * tan(∠POB), from the tangent value table, we find tan(30°) = √3/3.

4. Calculating, we get OD = OB = BP * tan(30°) = 6 * √3/3 = 2√3.

##5. Since OA and OB are the radii of circle O, OA = OB, according to the definition of an isosceles triangle, △OAB is an isosceles triangle. Since ∠POB = 60°, according to the definition of supplementary angles, we get ∠AOB = 120°. According to the properties of an isosceles triangle, in the isosceles triangle △OAB, ∠OAB = ∠OBA = 30°##.

6. Given OD ⊥ AB, thus ∠OCB = 90°, ##according to the triangle angle sum theorem, we get ∠COB = 180° - ∠OBC - ∠OCB = 180° - 30° - 90° = 60°##.

##7. According to the properties of a 30°-60°-90° triangle, in △OBC, OC is half of OB, thus OC = 1/2 * OB = 1/2 * 2√3 = √3##.

##8. Therefore, CD = OD - OC = 2√3 - √3 = √3##.

圆; 垂线; 切线; 直角三角形; 正弦

Property of the Tangent Line to a Circle

A tangent line to a circle is perpendicular to the radius that passes through the point of tangency.

In the figure of this problem, in circle O, point B is the point of tangency between line PB and the circle, <note>line segment OB is the radius of the circle</note>. According to the property of the tangent line to a circle, <note>the tangent line PB is perpendicular to the radius OB at the point of tangency B</note>, that is, <note>∠OBP=90 degrees</note>.

Triangle Angle Sum Theorem

<note>In triangle POB, angle OPB, angle OBP, and angle POB are the three interior angles of triangle POB</note>, according to the Triangle Angle Sum Theorem, <note>angle OPB + angle OBP + angle POB = 180°</note>, <note>in triangle OCB, angle OCB, angle CBO, and angle COB are the three interior angles of triangle OCB</note>, according to the Triangle Angle Sum Theorem, <note>angle OCB + angle CBO + angle COB = 180°</note>, <note>in triangle OAB, angle OAB, angle ABO, and angle AOB are the three interior angles of triangle OAB</note>, according to the Triangle Angle Sum Theorem, <note>angle OAB + angle ABO + angle OBA = 180°</note>.

Definition of Right Triangle

<note>In the figure of this problem, in triangle OPB, angle OBP is a right angle (90 degrees), so triangle OPB is a right triangle. Side OB and side BP are the legs, and side OP is the hypotenuse.</note>
<note>In triangle OCB, angle OCB is a right angle (90 degrees), so triangle OCB is a right triangle. Side OC and side BC are the legs, and side OB is the hypotenuse.</note>

Definition of Radius

A radius is a line segment joining the center of a circle to any point on the circumference of the circle.

In the figure of this problem, <note>in circle O, point O is the center of the circle</note>, <note>point A and point B are any points on the circle</note>, <note>line segment OA and line segment OB are line segments from the center to any point on the circle</note>, therefore <note>line segment OA and line segment OB are the radii of the circle</note>.

Definition of Tangent Function

In a right-angled triangle, the tangent value of an acute angle is equal to the length of the opposite side divided by the length of the adjacent side.

<note>In the right triangle POB</note>, <note>angle ∠POB is an acute angle</note>, <note>side PB is the opposite leg of angle ∠POB</note>, <note>side OB is the adjacent leg of angle ∠POB</note>, therefore <note>the tangent value of angle ∠POB is equal to the length of side PB divided by the length of side OB</note>, i.e., <note>tan(∠POB) = PB / OB</note>.

Definition of Isosceles Triangle

<note>Sides AO and BO are equal</note>, therefore <note>triangle AOB is an isosceles triangle</note>.

Properties of Isosceles Triangle

<note>In the isosceles triangle AOB, sides AO and OB are equal</note>. Therefore, according to the <note>properties of isosceles triangles</note>, the angles opposite the equal sides are equal, i.e., <note>angle OAB = angle OBA</note>.

Definition of Linear Pair of Angles

Two angles are in a linear pair if they share a common side, and their other sides are extensions of one another in opposite directions.

Properties of 30°-60°-90° Triangle

In a 30°-60°-90° triangle, the side opposite the 30-degree angle (the shorter leg) is half the length of the hypotenuse. The side opposite the 60-degree angle (the longer leg) is √3 times the length of the shorter leg.

In the <note>30°-60°-90° triangle OCB</note>, <note>angle OBC is 30 degrees</note>, <note>angle BOC is 60 degrees</note>, <note>angle OCB is 90 degrees</note>. <note>Side OB is the hypotenuse</note>, <note>side OC is the side opposite the 30-degree angle</note>, <note>side CB is the side opposite the 60-degree angle</note>. According to the properties of a 30°-60°-90° triangle, <note>side OC is equal to half of side OB</note>, <note>side CB is equal to √3 times side OC</note>. That is: <note>OC = 1/2 * OB</note>, <note>CB = OC * √3</note>.


Question: Find x.

Answer: 2

1. From the figure, it can be obtained: two chords AB and CD intersect at point E, and AE=x, CE=3, ED=4, EB=6.

2. According to the intersecting chords theorem, the theorem states: If two chords AB and CD intersect at point E, then AE * EB=CE * ED.

3. Substitute the known conditions into the theorem to obtain the equation x * 6=3 * 4.

4. Calculate the result of the equation to get 6x=12.

6. Through the above reasoning, the final answer is x=2.

Definition of Chord

A chord is defined as a line segment that connects any two points on a circle.

In the figure of this problem, in the circle, <note>point A</note> and <note>point B</note> are any two points on the circle, <note>line segment AB</note> connects these two points, so <note>line segment AB</note> is a chord of the circle. Similarly, <note>point C</note> and point D are any two points on the circle, <note>line segment CD</note> connects these two points, so <note>line segment CD</note> is a chord of the circle.

Intersecting Chords Theorem

If two chords AB and CD intersect at point E, then AE * EB = CE * ED.

In the figure of this problem, two <note>chords AB and CD</note> intersect at point E inside the circle. <note>Chord AB</note> is divided into segments AE and <note>EB</note> by point E, and <note>chord CD</note> is divided into segments <note>CE and ED</note> by point E. According to the Intersecting Chords Theorem, the product of the lengths of segment AE and segment EB is equal to the product of the lengths of segment CE and segment ED. Given AE=x, CE=3, <note>ED</note>=4, and <note>EB</note>=6, the equation derived from the theorem is x * 6 = 3 * 4.


Question: Find x.

Answer: 3.5

1. Given that ##point W, point W' and point F## are on the same line, according to the figure, ##segment WW' = x and segment W'F = 10.5## are the two parts near point F, with a total length of 14.

2. According to the calculation rule of segment addition, ##WF = W'W + W'F = x + 10.5 = 14##.

####

##3##. It is calculated that WW' = 3.5.

####

##4##. From the above reasoning steps, it can be concluded that the final answer is x = 3.5.

Definition of Line Segment

A line segment is a portion of a straight line that includes two endpoints and all the points between them.

In the figure of this problem, <note>line segment WF</note> is a part of a straight line, including <note>endpoint W and endpoint F</note> and all points between them, with a length of ##14##. <note>line segment WW'</note> is a part of a straight line, including <note>endpoint W and endpoint W'</note> and all points between them, with a length of x. <note>line segment W'F</note> is a part of a straight line, including <note>endpoint W' and endpoint F</note> and all points between them, with a length of ##10.5##.


Question: In \odot M, F L = 24, H J = 48, and m \widehat H P = 65. Find F G.

Answer: 48

2. ##Points F, G, K are on circle M, where KM is the radius of circle M and FG is the chord of circle M. Given KM⊥FG, according to the perpendicular bisector theorem, a diameter perpendicular to a chord will bisect the chord and the arcs subtended by the chord##.

3. ##Given KM bisects FG and KM intersects FG at point L, then L is the midpoint of FG, so FL=GL=(1/2)FG. Given FL=24, so FG=2FL=2*24=48##.

4. ##Through the above reasoning, the final answer is ##FG##=48.

圆; 弦; 圆心角; 垂直平分线; 直角三角形

Definition of Circle

A circle is the locus of all points in a plane that are at a fixed distance from a given point.

In <note>circle M</note>, <note>point M is the center</note>, and the radius is <note>KM</note>. All points in the figure that are at a distance of <note>KM from point M</note> are on <note>circle M</note>.

Definition of Radius

<note>In the diagram for this problem, in circle M, point M is the center of the circle, point K is any point on the circle, and segment KM is the segment from the center to any point on the circle, therefore segment KM is the radius of the circle.</note>

Perpendicular Diameter Theorem

If a diameter is perpendicular to a chord, then that diameter bisects the chord and also bisects the two arcs subtended by that chord.

<note>Radius KM is perpendicular to chord FG</note>, then according to the Perpendicular Diameter Theorem, <note>radius KM bisects chord FG, that is FL=GL</note>, and <note>diameter KM bisects the two arcs subtended by chord FG, that is arc FK=arc GK</note>.


Question: Find the perimeter of the parallelogram. If necessary, round to the nearest tenth.

Answer: 50

1. Given that the lengths of two pairs of adjacent sides of a parallelogram are 12 cm and 13 cm respectively. According to the ###properties theorem of parallelogram##, the opposite sides are equal, that is, the lengths of the other pair of sides are also 12 cm and 13 cm.

2. The perimeter of the parallelogram is equal to the sum of the lengths of its four sides. Therefore, the perimeter of the parallelogram is 12 cm + 13 cm + 12 cm + 13 cm.

3. By calculation, the perimeter of the parallelogram is 12 + 13 + 12 + 13 = 50 cm.

4. Each step in the above calculation is reasonable####, and the final solution gives the perimeter of the parallelogram.

Definition of Parallelogram

<note>The two pairs of opposite sides of the parallelogram are 12 cm and 13 cm respectively</note>. According to the definition, <note>these two pairs of opposite sides are parallel and equal</note>, thus forming a parallelogram.

Properties of Parallelogram Theorem

In a parallelogram, opposite angles are equal, opposite sides are equal, and the diagonals bisect each other.

In the figure of this problem, it is known that one pair of adjacent sides of the parallelogram are <note>12 centimeters and 13 centimeters</note>. According to the <note>Properties of Parallelogram Theorem</note>, the other pair of sides are also <note>12 centimeters and 13 centimeters</note>.

Perimeter of a Parallelogram

The perimeter of a parallelogram is equal to twice the sum of the lengths of its two adjacent sides. The formula is: \( P = 2(a + b) \), where \( a \) and \( b \) are the lengths of the two adjacent sides of the parallelogram.

In the diagram of this problem, in the parallelogram, <note>side 12cm and side 13cm are adjacent sides</note>, <note>side 12cm and side 13cm are also adjacent sides</note>. According to the formula for the perimeter of a parallelogram, the perimeter of the parallelogram is equal to twice the sum of its two pairs of adjacent sides, that is, <note>perimeter P = 2(12 + 13) = 50cm</note>.


Question: △ J K L ∼ △ W Y Z. Find x.

Answer: 21

1. The given condition is ΔJKL is similar to ΔWYZ. According to the definition of similar triangles, the corresponding angles of similar triangles are equal.

2. From the figure, it can be seen that ∠J corresponds to ∠W, ∠K corresponds to ∠Z, and ∠L corresponds to ∠Y.

3. Since ∠W = 71°, and ∠J = (4x - 13)°, therefore, according to the equality of corresponding angles in similar triangles, we have ∠J = ∠W.

4. From step 3, we obtain the equation (4x - 13)° = 71°.

5. Solving the equation (4x - 13) = 71 gives 4x = 84, x = 21.

6. Therefore, the value of x is 21.

Definition of Similar Triangles

Two triangles are similar if and only if their corresponding angles are equal and their corresponding sides are proportional.

In the figure of this problem, <note>ΔJKL and ΔWYZ are similar</note>, therefore according to the definition of similar triangles: <note>∠J = ∠W, ∠K = ∠Y, ∠L = ∠Z</note>; <note>JK/WY = KL/ZY = JL/WZ</note>.


Question: Find the area of the figure, accurate to one decimal place.

Answer: 31.1

1. The figure consists of a rectangle in the middle and two semicircles at both ends. The long side of the rectangle is 8, and the short side is 3.

2. According to the area formula of a rectangle in plane geometry, the area of the rectangle is equal to the long side multiplied by the short side, so the area of the rectangle is 8 × 3 = 24 square units.

3. The two semicircles combine into a complete circle, and the diameter of the circle is equal to the length of the short side of the rectangle, which is 3.

4. The radius of the circle is half of the diameter, which is 3 ÷ 2 = 1.5.

5. According to the area formula of a circle ##A = πr?##, where r is the radius, the area of the circle is ##π × (1.5)?## = 2.25π.

6. Therefore, the total area of the entire figure is the area of the rectangle plus the area of the two semicircles (i.e., a complete circle), which is 24 + 2.25π.

7. Calculate 24 + 2.25π ≈ 24 + 7.07 ≈ 31.07.

8. The final area of the figure is 31.07, rounded to the nearest tenth is 31.1.

Definition of Rectangle

A quadrilateral is called a rectangle if and only if each of its interior angles is a right angle (90°) and its opposite sides are both parallel (∥) and congruent (≅).

<note>In the figure of this problem, the middle part of the shape is a rectangle with a length of 8 and a width of 3.</note> The four interior angles of the rectangle are <note>right angles (90 degrees)</note>, and the opposite sides are parallel and equal in length. Specifically, <note>the long side of the rectangle is 8 and the short side is 3.</note>

Definition of Circle

<note>The centers of the two semicircles are located at the midpoints of the short sides of the rectangle</note>, <note>with a radius of 1.5</note>. In the figure, <note>all points that are at a distance of 1.5 from these centers are on the corresponding semicircle</note>.

Definition of Diameter

A diameter is a line segment that passes through the center of a circle and has its endpoints on the circle. It is the longest chord of the circle, with a length equal to twice the radius of the circle.

<note>In the figure of this problem, the diameter of the semicircle at both ends of the figure is the short side of the figure, which is 3 units</note>, <note>connecting the center of the circle and two points on the circumference</note>, the length is twice the radius, that is, <note>diameter = 2 × radius</note>.

Definition of Radius

<note>Each semicircle's radius</note> is <note>half of the rectangle's shorter side (3 units)</note>, which is 1.5 units. According to the <note>definition of radius</note>, <note>the radius is the line segment from the center of the circle to any point on the circle</note>, so in this problem's diagram, <note>the radius of the semicircle is 1.5 units</note>.

Formula for the Area of a Rectangle

The area of a rectangle is equal to its length multiplied by its width.

The length of the rectangle is <note>8</note>, the width is <note>3</note>, so the area of the rectangle = <note>8 * 3 = 24 square units</note>.

Area Formula of a Circle

The area of a circle is given by the formula A = πr², where A represents the area, π (pi) is the mathematical constant approximately equal to 3.14159, and r is the radius of the circle.

In the figure of this problem, since <note>two semicircles merge into a complete circle</note>, <note>the diameter of the circle is 3, so the radius is 1.5</note>. According to the area formula of a circle, the area A of the circle is equal to the circumference ratio <note>π multiplied by the square of the radius 1.5</note>, that is, <note>A = π(1.5)^2 = 2.25π</note>.

Definition of Semicircle

A semicircle is a geometric figure constructed from a diameter and an arc of a circle, meaning it represents one of the two congruent parts into which a circle is divided by its diameter.

<note>The two ends of the semicircle are respectively composed of a diameter and an arc</note>.


Question: Find x.

Answer: 93

1. Let the triangle be ABC, the given conditions are ∠A = 52°, ∠B = 35°, ∠C = x°.

2. According to the triangle angle sum theorem, ∠A + ∠B + ∠C = 180°.

3. Substituting the given conditions, we get 52° + 35° + x° = 180°.

4. By calculation, x° = 180° - 52° - 35°.

6. Through the above reasoning, the final answer is x = 93°.

普通三角形; 三角形的外角

Triangle Angle Sum Theorem

The sum of the interior angles of any triangle is 180°.

In the figure of this problem, <note>the three interior angles of the triangle are 52°, 35°, and x°</note>. According to the <note>Triangle Angle Sum Theorem</note>, <note>52° + 35° + x° = 180°</note>.


Question: Find x such that m \parallel n.

Answer: 20

1. Given lines m and n, and an angle formed by a transversal (4x - 23)° and (2x + 17)°. Let line m be AB, line n be CD, and the transversal be EF. The transversal EF intersects AB at M and intersects CD at N.

2. According to the parallel postulate 2: If two parallel lines are cut by a third line, corresponding angles are equal.

3. According to the definition of vertical angles, the vertical angle of (4x - 23)° is ∠FMB, so (4x - 23)° = ∠FMB. According to the definition of corresponding angles, ∠FMB and (2x + 17)° are corresponding angles because they are on the same side of the transversal and between the two lines.

4. According to the parallel postulate 2, if m || n, then ∠FMB = (2x + 17)°. Given (4x - 23)° = ∠FMB, so (4x - 23)° = (2x + 17)°.

5. Solve the equation: 4x - 23 = 2x + 17.

6. Move 2x to the left side of the equation and move -23 to the right side of the equation, resulting in: 4x - 2x = 17 + 23.

9. Through the above reasoning, if m ∥ n, then x = 20.

Definition of Parallel Lines

Two lines in the same plane that do not intersect are called parallel lines.

<note>Line m and line n lie in the same plane</note>, and <note>they have no intersection points</note>, therefore according to the definition of parallel lines, <note>line m and line n are parallel lines</note>.

Definition of Vertical Angles

Vertical angles are the pair of opposite angles formed when two lines intersect. These angles are equal in measure.

The original text: <note>Two intersecting lines EF and AB intersect at point M</note>, forming four angles: <note>angle EMA</note>, <note>angle FMB</note>, <note>angle EMB</note>, and <note>angle AMF</note>. According to the definition of vertical angles, <note>angle EMA and angle FMB are vertical angles</note>, <note>angle EMB and angle AMF are vertical angles</note>. Since vertical angles are equal, <note>angle EMA = angle FMB</note>, <note>angle EMB = angle AMF</note>.

Parallel Postulate 2 of Parallel Lines

If two parallel lines are cut by a transversal, then the corresponding angles are equal, the alternate interior angles are equal, and the consecutive interior angles on the same side of the transversal are supplementary.

In the figure of this problem, <note>two parallel lines AB and CD are intersected by a third line EF</note>, forming the following geometric relationship: <note>corresponding angles: angle FMB and angle FND are equal</note>. These relationships illustrate that <note>when two parallel lines are intersected by a third line, the corresponding angles are equal</note>.

Definition of Corresponding Angles

Two angles are referred to as corresponding angles if they are formed when two lines, a and b, are intersected by a third line, c, and both angles lie on the same side of the intersecting line c and on the same side of the lines a and b.

<note>Two parallel lines AB and CD</note> are intersected by <note>a line EF</note>, where <note>angle FMB and angle FND are on the same side of the intersecting line EF</note>, on the <note>same side of the two intersected lines AB and CD</note>. Therefore, <note>angle FMB and angle FND are corresponding angles</note>. <note>Corresponding angles are equal</note>, that is, <note>angle FMB is equal to angle FND</note>.


Question: Find x. Round to the nearest tenth.

Answer: 17.4

1. Given the conditions for triangle NOP, where ## ∠NPO## = 38°, NP = 20√2, NO = x, and ∠NOP is a right angle, ## according to the definition of a right triangle, since ∠NOP = 90°, triangle NOP is a right triangle ##.

2. ## According to the definition of the sine function, sin(∠NPO) = opposite side / hypotenuse ##, where the opposite side is NO and the hypotenuse is NP.

3. According to the definition of the sine function, sin(38°) = NO / NP.

4. Substituting the given conditions, we get sin(38°) = x / (20√2).

5. Look up and calculate the value of sin(38°) (sin(38°) ≈ 0.6157), and substitute it into the equation: 0.6157 = x / (20√2).

6. Through algebraic operations, solve for the value of x, i.e., x = 0.6157 × 20√2.

7. Calculate 0.6157 × 20√2 ≈ 0.6157 × 28.2843 ≈ ##17.415##.

8. Round the result to the nearest tenth, obtaining x ≈ 17.4.

9. Through the above reasoning, the final answer is 17.4.

直角三角形; 正弦; 余弦

Definition of Sine Function

In a right-angled triangle, the sine of an acute angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

In the figure of this problem, in the right triangle NOP, <note>angle ∠NPO is an acute angle</note>, <note>side NO is the opposite side of angle ∠NPO</note>, <note>side NP is the hypotenuse</note>. According to the definition of the sine function, <note>the sine value of angle ∠NPO is equal to the ratio of the opposite side NO to the hypotenuse NP</note>, that is, sin(38°) = NO / NP.

Definition of Right Triangle

A triangle is classified as a right triangle if it has one interior angle measuring exactly 90 degrees.

<note>Angle ∠NOP is a right angle (90 degrees)</note>, therefore triangle NOP is a right triangle. Side NO and side PO are the legs, and side NP is the hypotenuse.


Question: As shown in the figure, PA and PB are tangent to ⊙O at A and B respectively. Points C and D are moving points on segments PA and PB respectively, and CD always remains tangent to circle O. If PA=8cm, then the perimeter of △PCD is ()

Answer: 16

1. Given PA and PB are tangents to ⊙O at A and B respectively, and PA=8 cm, ##according to the tangent length theorem,## PB=8 cm.

2. Connect CD, and CD is tangent to ⊙O at point E.

3. Since CD is tangent to ⊙O at point E, according to ##the tangent length theorem##, CA=CE, DE=DB.

4. Since point C is on PA and point D is on PB, the perimeter of △PCD is PC+PD+CD.

5. Because CA=CE and DE=DB, CD=CE+DE##=CA+DB##.

6. Based on the above equalities, we can deduce the perimeter of △PCD=PC+PD+##CA+CD##=PA+PB=8 cm + 8 cm.

7. Through the above reasoning, the final answer is 16 cm.

圆; 切线; 等腰三角形; 线段

Definition of Tangent to a Circle

A line is a tangent to a circle if and only if it has exactly one point of intersection with the circle. This point of intersection is called the point of tangency.

Original text: <note>Circle O and line PA have only one common point A, which is called the point of tangency</note>. Therefore, <note>line PA is the tangent to circle O</note>. Similarly, <note>circle O and line PB have only one common point B, which is called the point of tangency</note>. Therefore, <note>line PB is the tangent to circle O</note>. <note>Circle O and line CD have only one common point E, which is called the point of tangency</note>. Therefore, <note>line CD is the tangent to circle O</note>.

Tangent-Segment Theorem

If two tangent segments are drawn to a circle from an external point, then the lengths of these tangent segments are equal. Moreover, the line segment connecting the center of the circle to the external point bisects the angle formed by the two tangent segments.

In the figure of this problem, from a point outside the circle <note>C</note>, two tangents <note>CA</note> and <note>CE</note> are drawn to the circle, and their tangent lengths are equal, i.e., <note>CA=CE</note>; similarly, from a point outside the circle <note>D</note>, two tangents <note>DB</note> and <note>DE</note> are drawn to the circle, and their tangent lengths are equal, i.e., <note>DB=DE</note>; similarly, from a point outside the circle <note>P</note>, two tangents <note>PA</note> and <note>PB</note> are drawn to the circle, and their tangent lengths are equal, i.e., <note>PA=PB</note>.

Formula for the Perimeter of a Triangle

The perimeter of a triangle is the sum of the lengths of its three sides. Therefore, if the three sides of the triangle are denoted as \( a \), \( b \), and \( c \), then the perimeter \( P \) is given by \( P = a + b + c \).

<note>Given that the three sides of triangle PCD are PC, PD, and CD</note>, according to the formula for the perimeter of a triangle, <note>the perimeter of the triangle is equal to the sum of the lengths of its three sides</note>, that is, <note>perimeter P=PC+PD+CD</note>.


Question: Find m ∠ W.

Answer: 108

1. From the figure, in quadrilateral XYZW, angle X = x°, angle Y = 2x°, angle W = 3x°, angle Z = 4x°.

2. According to the quadrilateral interior angle sum theorem, the sum of the four interior angles of a quadrilateral is 360°, thus angle X + angle Y + angle Z + angle W = x° + 2x° + 3x° + 4x° = 360°.

3. Simplifying the equation, we get 10x° = 360°.

4. Solving the equation, we find x = 36°.

5. Knowing angle W = 3x°, substituting x = 36°, we get angle W = 3 * 36° = 108°.

6. Through the above reasoning, the final answer is 108°.

Definition of Interior Angles of a Polygon

The interior angles of a polygon are the angles formed between each pair of adjacent sides within the polygon, specifically referring to the angles inside the polygon.

In the figure of this problem, the interior angles of quadrilateral XYZW are <note>∠WXY</note>, <note>∠XYZ</note>, <note>##∠XWZ##</note>, and <note>##∠YZW##</note>, corresponding to <note>x°</note>, <note>2x°</note>, <note>3x°</note>, and <note>4x°</note> respectively.

Sum of Interior Angles of a Quadrilateral Theorem

In any quadrilateral, the sum of the four interior angles is 360°.

<note>The four interior angles ∠X, ∠Y, ∠W, and ∠Z of quadrilateral XYZW</note> satisfy <note>∠X + ∠Y + ∠W + ∠Z = 360°</note>.

Definition of Quadrilateral

A quadrilateral is a closed geometric figure composed of four line segments, which are referred to as the sides of the quadrilateral. It has four vertices and four interior angles.

In the figure of this problem, <note>quadrilateral XYZW is composed of four line segments XY, WZ, XW, and YZ</note>, which are called the sides of the quadrilateral. <note>Quadrilateral XYZW has four vertices, namely point X, point Y, point W, and point Z</note>, and <note>has four interior angles, namely angle XWZ, angle XYZ, angle YZW, and angle YXW</note>.


Question: If S T = 8, T R = 4, and P T = 6, find Q R.

Answer: 9

1. From the figure, PT∥QR. According to the parallel axiom 2 of parallel lines, corresponding angles are equal, i.e., ∠SPT = ∠SQR.

2. Also, since angle S is the common angle of △QSR and △PST, and ∠SPT = ∠SQR, according to the AA criterion for similar triangles, △QSR∽△PST.

3. Because △QSR∽△PST, according to the definition of similar triangles, ST/SR=PT/QR.

4. Given ST = 8, TR = 4, and PT = 6, substituting into the above proportion gives 8/(8+4)=6/QR=8/12=6/QR, i.e., QR=6 * 12/8=9.

等腰三角形; 平行线; 普通三角形

Parallel Postulate 2 of Parallel Lines

<note>In the figure of this problem, two parallel lines PT and QR are intersected by a third line SQ</note>, forming the following geometric relationship: <note>corresponding angles: angle SPT and angle SQR are equal</note>.

Definition of Similar Triangles

Two triangles are similar if and only if their corresponding angles are equal and their corresponding sides are proportional.

In the figure of this problem, <note>△QSR and △PST are similar triangles</note>. According to the definition of similar triangles: <note>∠QSR = ∠PST</note>, <note>∠SPT = ∠SQR</note>, <note>∠STP = ∠SRQ</note>; <note>ST/SR=PT/QR=SP/SQ</note>.

Similarity Theorem for Triangles (AA)

Two triangles are similar if two of their corresponding angles are congruent.

The original text: <note>Angle QSR is equal to angle PST</note>, and <note>angle SPT is equal to angle SQR</note>, so <note>triangle QSR is similar to triangle PST</note>.

Definition of Corresponding Angles

<note>Two parallel lines PT and QR</note> are intersected by a <note>line SQ</note>, where <note>angle SPT</note> and <note>angle SQR</note> are located on <note>the same side of the intersecting line SQ</note>, on <note>the same side of the two intersected lines PT and QR</note>. Therefore, <note>angle SPT and angle SQR are corresponding angles</note>. <note>Corresponding angles are equal</note>, that is, <note>angle SPT is equal to angle SQR</note>.

Definition of Parallel Lines

<note>Line PT and line QR</note> are located in the same plane and they do not intersect, so according to the definition of parallel lines, <note>line PT and line QR</note> are parallel lines.


Question: m \widehat A C = 160 和 m \angle B E C = 38。求 m \angle A E B?

Answer: 42

1. The inscribed angle theorem states that an inscribed angle is equal to half of the central angle that subtends the same arc. The problem gives m \\widehat{AC} = 160.

2. According to the inscribed angle theorem, the inscribed angle \\angle AEC subtends arc AC, and the corresponding central angle is m \\angle AEC = \\frac{1}{2} \\times 160 = 80.

3. Then, another angle given in the figure is m \\angle BEC = 38.

4. Therefore, \\angle AEB = \\angle AEC - \\angle BEC.

5. Substituting the above results, m \\angle AEB = 80 - 38.

6. Through the above reasoning, the final answer is 42.

Inscribed Angle Theorem

In a circle, an inscribed angle is equal to half of the central angle that subtends the same arc.

In the figure of this problem, <note>\(\angle A E C\) is an inscribed angle</note>, corresponding to the arc <note>\(\widehat{AC}\)</note>. According to the problem statement, <note>\(m\widehat{AC} = 160\) degrees</note>. According to the Inscribed Angle Theorem, <note>\(\angle A E C = \frac{1}{2} \times 160 = 80\) degrees</note>.

Definition of Central Angle

An angle formed by two radii connecting two points on the circumference of a circle to the center of the circle is called a central angle.

<note>In the figure of this problem, in the circle, point A and point C are two points on the circle, and the center of the circle is point O</note>. <note>The angle ∠AOC formed by the lines OA and OC is called the central angle</note>.

Definition of Inscribed Angle

An inscribed angle is an angle whose vertex lies on the circumference of a circle and whose sides intersect the circle at two points.

In circle O, the vertex of angle AEC (point E) is on the circumference, and the two sides of angle AEC intersect circle O at points A and C respectively. Therefore, angle AEC is an inscribed angle.


Question: Use parallelogram M N P R to find x.

Answer: 8

1. Given parallelogram M N P R, by definition, opposite sides are equal, therefore M N = P R.

2. Let M N = 3x - 4, P R = 20, according to the properties of the parallelogram, 3x - 4 = 20.

3. Solve the equation 3x - 4 = 20, obtaining 3x = 24, then x = 24 / 3 = 8.

4. Through the above reasoning, we conclude x = 8.

平行四边形; 平行线; 对顶角; 内错角; 同旁内角

Definition of Parallelogram

A quadrilateral is a parallelogram if and only if it has two pairs of opposite sides that are both parallel (∥) and congruent (≅).

In the figure of this problem, the quadrilateral M N P R is a parallelogram, <note>side M N is parallel and equal to side P R</note>, <note>side M R is parallel and equal to side N P</note>.

Properties of Parallelogram Theorem

In a parallelogram, opposite angles are equal, opposite sides are equal, and the diagonals bisect each other.

In the diagram of this problem, in the parallelogram <note>M N P R</note>, sides <note>M N and P R are equal</note>, sides <note>M R and N P are equal</note>; <note>diagonals M P and N R bisect each other</note>, that is, the intersection point Q divides the diagonal <note>M P into two equal segments M Q and Q P</note>, and divides the diagonal <note>N R into two equal segments N Q and Q R</note>. Given <note>M N = 3x - 4 and P R = 20</note>, we get <note>3x - 4 = 20</note>.


Question: Find w in the given parallelogram.

Answer: 5

1. Given that quadrilateral FGHJ is a parallelogram, ##let the two diagonals FH and GJ intersect at point K##. According to the ##properties theorem## of parallelograms, the diagonals of a parallelogram bisect each other, i.e., FK = ##KH##.

2. ##From the figure, we have FK=4w-7, KH=2w+3. Substituting these into the equation, we get 4w - 7 = 2w + 3##.

3. Solving the equation 4w - 7 = 2w + 3, we get: 2w = 10, hence w = 5.

4. Through the above reasoning, the final answer is 5.

平行四边形; 对顶角; 内错角

Definition of Parallelogram

<note>Quadrilateral FGHJ is a parallelogram</note>, <note>side FG is parallel and equal to side JH</note>, <note>side FJ is parallel and equal to side GH</note>.

Properties of Parallelogram Theorem

In the diagram of this problem, in the parallelogram FGHJ, <note>angles ∠FGH and ∠HJF</note> are equal, <note>angles ∠GFJ and ∠JHG</note> are equal; <note>sides FG and JH</note> are equal, <note>sides FJ and GH</note> are equal; <note>diagonals FH and GJ</note> bisect each other, that is, the intersection point K divides <note>diagonal FH</note> into two equal segments FK and <note>KH</note>, and divides <note>diagonal JG</note> into two equal segments JK and <note>KG</note>.


Question: Find the measure of m \\angle 1.

Answer: 109

1. According to the properties of the central angle and the definition of the central angle, let the central angle corresponding to the 218° arc be ∠2, i.e., ∠2 = 218°.

2. According to the inscribed angle theorem: In a circle, the inscribed angle is equal to half of the central angle corresponding to the arc it intercepts.

3. The inscribed angle (∠1) is equal to half of the corresponding central angle (218°).

5. Through reasoning, it is finally concluded that the measure of ∠1 is 109°.

Inscribed Angle Theorem

The arc measured by the <note>inscribed angle ∠1</note> is 218°. According to the <note>Inscribed Angle Theorem</note>, the inscribed angle (i.e., <note>∠1</note>) is equal to half of the corresponding <note>central angle (i.e., 218°)</note>, that is, <note>m ∠1 = 218° ÷ 2 = 109°</note>.

Definition of Inscribed Angle

In the circle, <note>the vertex of angle 1 is on the circumference</note>, <note>the two sides of angle 1 intersect the circle at two points</note>. Therefore, <note>angle 1 is an inscribed angle</note>.

Definition of Central Angle

<note>The angle ∠2 formed by connecting the center of the circle and two points on the circle is called the central angle</note>.

Properties of Central Angles

The measure of a central angle is equal to the measure of the arc that it intercepts.

In the figure of this problem, it is known that the <note>central angle ∠2</note> corresponds to an <note>arc measure of 218°</note>. According to the <note>properties of central angles</note>, the <note>measure of a central angle is equal to the measure of its corresponding arc</note>, that is, <note>the measure of ∠2 = 218°</note>.


Question: Find the measure of JK.

Answer: 31

2. Triangles JNK and JNL share a right angle side JN, so according to the congruent triangles theorem (SAS), triangle JNK is congruent to triangle JNL, therefore JK=JL, thus we have 6x + 7 = 9x - 5.

3. Solving the equation 6x + 7 = 9x - 5, first subtract 6x from both sides to get 7 = 3x - 5.

6. Substitute x = 4 into the original expression 6x + 7, we get 6*4 + 7 = 24 + 7 = 31.

7. Therefore, the measurement of JK is 31.

8. Through the above reasoning, the final answer is 31.

等腰三角形; 中点; 线段

Triangular Congruence Theorem (SAS)

If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.

In this problem diagram, in <note>triangle JNK and triangle JNL</note>, <note>side JN is the common side of the two triangles</note>, <note>side KN is equal to side NL</note>, and <note>angle JNL is equal to angle JNK</note>. Therefore, according to the Triangular Congruence Theorem (SAS), these <note>two triangles are congruent</note>.

Definition of Congruent Triangles

Two triangles are congruent if and only if their corresponding sides are equal and their corresponding angles are equal.

<note>Triangles JNK and JNL are congruent triangles</note>, <note>the corresponding sides and corresponding angles of triangle JNK are equal to those of triangle JNL</note>, that is:
<note>side JN = side JN</note>
<note>side JK = side JL</note>
<note>side KN = side LN</note>
At the same time, the corresponding angles are also equal:
<note>angle JNK = angle JNL</note>
<note>angle JKN = angle JKL</note>
<note>angle KJN = angle LJN</note>


Question: In triangle XYZ, P is the centroid, KP = 3, and XJ = 8. Find YJ.

Answer: 8

1. Given triangle XYZ, P is the centroid. According to the centroid theorem, ZJ is the median from vertex Z to side XY. By the definition of the median in a triangle, XJ = YJ.

2. Given XJ = 8, then YJ = XJ = 8.

3. Through the above reasoning, the final answer is YJ = 8.

普通三角形; 中点; 线段

Centroid Theorem

The centroid of a triangle is the point where its three medians intersect, and it divides each median into two segments, with the longer segment being twice the length of the shorter segment and connecting to the vertex.

<note>P is the centroid of triangle XYZ</note>. According to the Centroid Theorem, the centroid divides each median into two segments in a 2:1 ratio, with the longer segment connecting to the vertex. For example, in the <note>median XK</note>, <note>P divides it into two segments</note>, with the longer segment being <note>XP</note> and the shorter segment being <note>PK</note>. Similarly, in the <note>median ZJ</note> and <note>median YL</note>, <note>P also divides them into two segments in a 2:1 ratio</note>.

Definition of Median of a Triangle

A median of a triangle is a line segment drawn from one vertex of the triangle to the midpoint of the opposite side.

In <note>triangle XYZ</note>, <note>vertex Z</note> is a vertex of the triangle, and <note>opposite side XY</note> is the side opposite vertex Z. <note>Point J</note> is the midpoint of side XY, and <note>segment ZJ</note> is the segment from vertex Z to the midpoint J of the opposite side XY. Therefore, <note>ZJ</note> is a median of <note>triangle XYZ</note>.


Question: In the figure, m \\angle 1 = 50 and m \\angle 3 = 60. Find the measure of \\angle 4.

Answer: 50

1. Given conditions are ∠1 = 50°, ∠3 = 60°, j ∥ k, m ∥ n, p ∥ q.

2. According to the definition of vertical angles, let the other acute angle formed by the intersection of j and q be ∠a, then ∠1 and ∠a are vertical angles, ∠1 = ∠a = 50°; let the other acute angle formed by the intersection of p and k be ∠c, then ∠4 and ∠c are vertical angles, ∠4 = ∠c; according to the definition of corresponding angles, j ∥ k and intersect q at two points, let the acute angle on the upper right side formed by the intersection of k and q be ∠b, then ∠b and ∠a are corresponding angles, according to Parallel Postulate 2, ∠a = ∠b (corresponding angles are equal).

3. Given p ∥ q and intersect k at two points, according to the definition of alternate interior angles, ∠b and ∠c are alternate interior angles, and according to Parallel Postulate 2, ∠b = ∠c (alternate interior angles are equal).

4. Substituting the given conditions, ∠1 = ∠a = 50°, ∠a = ∠b, ∠b = ∠c, ∠4 = ∠c, so ∠4 = ∠c = ∠b = ∠a = ∠1 = 50°.

5. Through the above reasoning, the final answer is ∠4 = 50°.

同位角; 内错角; 平行线

Definition of Corresponding Angles

Two angles are referred to as corresponding angles if they are formed when two lines, a and b, are intersected by a third line, c, and both angles lie on the same side of the intersecting line c and on the same side of the lines a and b.

When <note>j ∥ k</note> and intersects with <note>q</note> forming <note>∠b and ∠a</note>, then <note>∠b and ∠a are corresponding angles</note>.

Definition of Alternate Interior Angles

When a straight line (referred to as a transversal) intersects two parallel lines, the angles that are located between the two parallel lines but on opposite sides of the transversal are called alternate interior angles.

<note>In the diagram of this problem, two parallel lines p and q are intersected by a transversal k</note>, <note>where ∠b and ∠c are located between the two parallel lines and on opposite sides of the transversal k, thus ∠b and ∠c are alternate interior angles</note>.

Parallel Postulate 2 of Parallel Lines

If two parallel lines are cut by a transversal, then the corresponding angles are equal, the alternate interior angles are equal, and the consecutive interior angles on the same side of the transversal are supplementary.

In the figure of this problem, two parallel lines <note>j and k</note> are intersected by a third <note>line q</note>, forming the following geometric relationship: corresponding angles: <note>angle a and angle b are equal</note>. Two parallel lines <note>p and q</note> are intersected by a third <note>line k</note>, forming the following geometric relationship: alternate interior angles: <note>angle c and angle b are equal</note>. These relationships illustrate that when two parallel lines are intersected by a third line, corresponding angles are equal, alternate interior angles are equal, and same-side interior angles are supplementary.

Definition of Vertical Angles

Vertical angles are the pair of opposite angles formed when two lines intersect. These angles are equal in measure.

In the diagram of this problem, two intersecting lines <note>j and k</note> intersect at a point, forming four angles. According to the definition of vertical angles, <note>angle 1 and angle a are vertical angles</note>. Since the angles of vertical angles are equal, <note>angle 1 = angle a</note>. Two intersecting lines <note>p and k</note> intersect at a point, forming four angles. According to the definition of vertical angles, <note>angle 4 and angle c are vertical angles</note>. Since the angles of vertical angles are equal, <note>angle 4 = angle c</note>.


Question: Find a in the given parallelogram.

Answer: 9

1. The given geometric figure is a parallelogram WXZY.

2. According to the properties theorem of parallelograms, the diagonals of a parallelogram bisect each other, so the intersection point of the diagonals is the midpoint of the diagonals.

3. Let the intersection point of the diagonals be point O, so OX=OY.

4. Given OY=2, so OX=2, i.e., a-7=2, solving: a=9.

5. Through the above reasoning, the final answer is a=9.

平行四边形; 平行线; 内错角

Definition of Parallelogram

In the figure of this problem, <note>parallelogram WXZY</note> is a parallelogram, <note>side WX is parallel and equal to side YZ</note>, <note>side WY is parallel and equal to side XZ</note>.

Properties of Parallelogram Theorem

In the parallelogram <note>WXZY</note>, the opposite angles <note>∠WXZ and ∠ZYW are equal</note>, the opposite angles <note>∠YWZ and ∠XZY are equal</note>; the sides <note>WX and YZ are equal</note>, the sides <note>WY and XZ are equal</note>; the diagonals <note>WZ and XY bisect each other, with the intersection point being O</note>, meaning the intersection point divides the diagonal <note>WZ into two equal segments WO and ZO</note>, and divides the diagonal <note>XY into two equal segments XO and YO</note>.


Question: Find the perimeter of the parallelogram. If necessary, round to the nearest tenth.

Answer: 44

1. ##Let the four vertices of the parallelogram be ABCD, then AD = 10 meters, AB = 12 meters.##

2. ##The perimeter of the parallelogram P = 2(a + b), that is, P = 2 * (AB + AD).##

3. ##Substitute into the equation to get, P=2 * (AB + AD) = 2 * (12 meters + 10 meters) = 2 * 22 meters.##

####

##4##. Calculate to get the perimeter of the parallelogram as 44 meters.

##5##. Therefore, the perimeter of the parallelogram is 44 meters.

平行四边形; 线段; 正弦

Perimeter of a Parallelogram

The perimeter of a parallelogram is equal to twice the sum of the lengths of its two adjacent sides. The formula is: \( P = 2(a + b) \), where \( a \) and \( b \) are the lengths of the two adjacent sides of the parallelogram.

<note>In the figure of this problem, in parallelogram ABCD, sides AB and AD are two adjacent sides</note>, according to the formula for the perimeter of a parallelogram, the perimeter of the parallelogram is equal to twice the sum of its two pairs of adjacent sides, that is, <note>Perimeter P = 2 *(AB + AD) = 2 *(12 meters + 10 meters) = 44 meters</note>.


Question: For trapezoid Q R T U, V and S are the midpoints of the two legs. If Q R = 4 and U T = 16, find V S.

Answer: 10

1. Given that quadrilateral QRTU is a trapezoid, QR is the upper base, UT is the lower base, and QR = 4, UT = 16.

2. According to the midsegment theorem of trapezoids, the length of the midsegment is equal to the average of the lengths of the upper base and the lower base.

3. In trapezoid QRTU, V and S are the midpoints of legs UQ and RT respectively.

4. Based on the above known conditions, VS is the midsegment of trapezoid QRTU.

5. According to the midsegment theorem of trapezoids, VS = (QR + UT) / 2.

6. Substituting the known conditions, VS = (4 + 16) / 2 = 20 / 2 = 10.

7. Through the above reasoning, the final answer is VS = 10.

Definition of Trapezoid

A quadrilateral is a trapezoid if and only if it has exactly one pair of parallel sides.

<note>In the figure of this problem, in quadrilateral QRTU, side QR and side UT are parallel</note>, while <note>side QU and side RT are not parallel</note>. Therefore, according to the definition of trapezoid, <note>quadrilateral QRTU is a trapezoid</note>, <note>because it has only one pair of parallel sides</note>.

Median Line Theorem of Trapezoid

The median line of a trapezoid is the line segment that connects the midpoints of the non-parallel sides. This line segment is parallel to the bases (the parallel sides of the trapezoid) and its length is equal to half the sum of the lengths of the two bases.

In the figure of this problem, in trapezoid QRTU, <note>side QR and side UT are the two bases of the trapezoid</note>, <note>point V and point S are the midpoints of the two legs of the trapezoid</note>, <note>line segment VS is the median line connecting the midpoints of the legs</note>. According to the Median Line Theorem of Trapezoid, <note>line segment VS is parallel to side QR and side UT</note>, and <note>the length of line segment VS is equal to half the sum of the lengths of side QR and side UT</note>, that is, <note>VS = (QR + UT) / 2</note>.


Question: Find x.

Answer: 12

1. Given that the triangle has a right angle, according to the definition of a right triangle, the triangle is a right triangle. The legs are 9 and x, and the hypotenuse is 15.

2. According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs, so we can set up the equation: 15^2 = 9^2 + x^2.

3. Calculate the value of 15^2 as 225, and the value of 9^2 as 81, thus the equation becomes 225 = 81 + x^2.

4. Move 81 to the other side of the equation, obtaining x^2 = 225 - 81.

5. Calculate the value of 225 - 81 as 144, thus obtaining x^2 = 144.

6. Take the square root of both sides of x^2 = 144, obtaining x = 12.

7. Through the above reasoning, the final answer is 12.

Definition of Right Triangle

A triangle is classified as a right triangle if it has one interior angle measuring exactly 90 degrees.

<note>This triangle is a right triangle</note>, in which <note>one interior angle is 90 degrees</note>. The legs are 9 and <note>x</note>, and the hypotenuse is <note>15</note>.

Pythagorean Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs.

In a <note>right-angled triangle</note>, the square of the hypotenuse is equal to the sum of the squares of the two legs. That is, <note>15^2 = 9^2 + x^2</note>.


Question: Find the value of the variable y in the figure.

Answer: 50

1. Let two horizontal lines A and B be intersected by an oblique line C. From the figure, it can be determined that lines A and B are parallel. According to the parallel axiom 2 of parallel lines, corresponding angles are equal, i.e., (3x - 15)° = 105°.

2. Solve the equation (3x - 15)° = 105°, eliminating the parentheses to get 3x - 15 = 105.

3. Add 15 to both sides of the equation to obtain 3x = 120.

4. Divide both sides by 3 to get x = 40.

5. Let the intersection point of line A and oblique line C be D, and the intersection point of line B and oblique line C be E. According to the definition of vertical angles, angle BED = 105°.

6. According to the parallel axiom 2 of parallel lines, consecutive interior angles are supplementary, satisfying (y + 25)° + angle BED = 180°.

7. Eliminating the parentheses gives y + 25 + 105 = 180.

8. Combine constants to get y + 130 = 180.

9. Subtract 130 from the left side of the equation to get y = 180 - 130.

11. Through the above reasoning, the final answer is y = 50.

内错角; 对顶角; 平行线

Parallel Postulate 2 of Parallel Lines

In the context of this problem, <note>two parallel lines are intersected by a third line</note>, forming the following geometric relationships: 1. <note>Corresponding angles</note>: the angle (3x - 15)° is equal to the angle 105°. 2. Consecutive interior angles: the angle (y + 25)° and <note>angle BED</note>° are supplementary, that is, (y + 25) + 105 = 180 degrees.

Definition of Corresponding Angles

In the figure of this problem, <note>two parallel lines A and B are intersected by a line C</note>, <note>where (3x - 15)° and 105° are on the same side of the intersecting line C, on the same side of the two intersected lines A and B</note>, therefore <note>(3x - 15)° and 105° are corresponding angles</note>. <note>Corresponding angles are equal, that is, (3x - 15)° is equal to 105°</note>.

Definition of Consecutive Interior Angles

When two lines are intersected by a third line, the angles that lie on the same side of the transversal and inside the two intersected lines are called consecutive interior angles. If the two lines are parallel, then the consecutive interior angles are supplementary. Conversely, if the consecutive interior angles are supplementary, then the two lines are parallel.

Original: <note>Two lines A and B are intersected by a third line C</note>, the angles <note>(y + 25)° and BED are on the same side of the transversal C</note>, and <note>within the lines A and B intersected by the transversal</note>, so <note>the angles (y + 25)° and BED</note> are consecutive interior angles. Consecutive interior angles <note>(y + 25)° and BED are supplementary</note>, that is, <note>(y + 25)° + BED = 180 degrees</note>.

Definition of Vertical Angles

<note>Two intersecting lines B and C intersect at point E</note>. According to the definition of vertical angles, <note>angle 105° and angle BED are vertical angles</note>. Since vertical angles are equal, <note>angle 105° = angle BED</note>.

Definition of Parallel Lines

Two lines in the same plane that do not intersect are called parallel lines.

<note>Line A and Line B are in the same plane</note>, and they <note>do not intersect</note>, so according to the definition of parallel lines, <note>Line A and Line B are parallel lines</note>.


Question: Express the value of \\cos P as a decimal rounded to two decimal places.

Answer: 0.47

##1##. To calculate the cosine value of angle P, the definition of cosine is: the length of the adjacent side divided by the length of the hypotenuse.

##2##. In triangle PQR, the adjacent side of angle P is PR = 8, and the hypotenuse is QR = 17.

##3##. Calculate the cosine value of angle P, i.e., cos P = adjacent side PR / hypotenuse QR = 8 / 17.

##4##. Calculate 8 / 17 ≈ 0.470588235, rounded to two decimal places is 0.47.

##5##. Through the above reasoning, the final answer is 0.47.

余弦; 直角三角形; 线段

Definition of Right Triangle

<note>Angle ∠PRQ is a right angle (90 degrees)</note>, therefore triangle PQR is a right triangle. <note>Side RQ</note> and <note>side PR are the legs</note>, <note>side QP is the hypotenuse</note>.

Cosine Function

The cosine function is a trigonometric function that can be defined via the sides of a right triangle. For an angle α, its cosine value is equal to the ratio of the length of the adjacent side to the hypotenuse. In the unit circle, the cosine value is the x-coordinate of the point on the circle that makes an angle α with the positive x-axis.

In the right triangle <note>PQR</note>, <note>side PR is the adjacent side of angle ∠PQR</note>, and <note>side QR is the hypotenuse</note>. Therefore, according to the definition of the cosine function, <note>the cosine value of angle ∠PQR</note> is equal to the ratio of the adjacent side PR to the hypotenuse QR, that is, <note>cos(∠PQR) = PR / QR = 8 / 17</note>.


Question: Find x.

Answer: 10

1. In the given triangle, one known angle is 30°. Another exterior angle is known to be a right angle. According to the triangle angle sum theorem, the angle opposite the side 10√3 should be 60°. It can be determined that this triangle is a special 30°-60°-90° triangle.

2. According to the properties of a 30°-60°-90° triangle, in a 30°-60°-90° triangle, the ratio of the hypotenuse to the short side is 2:1, and the ratio of the long side opposite the 60° angle to the short side is √3:1.

3. Suppose the short side is x. From the problem, it is known that the long side is 10√3.

4. Based on the ratio of the long side opposite the 60° angle to the short side in a 30°-60°-90° triangle, which is √3:1, we can establish the equation: 10√3 = x√3.

5. Solving the equation 10√3 = x√3, dividing both sides by √3, we get x = 10.

6. Through the above reasoning, the final answer is 10.

直角三角形; 正弦; 余弦

Definition of Triangle

A geometric figure is called a triangle if it is formed by three non-collinear points and the line segments connecting them.

<note>Let the triangle be triangle ABC, which is a geometric figure composed of three non-collinear points A, B, C and their connecting line segments AC, AB, BC</note>. <note>Points A, B, and C are the three vertices of the triangle</note>, <note>Line segments AC, AB, and BC are the three sides of the triangle</note>.

Triangle Angle Sum Theorem

The sum of the interior angles of any triangle is 180°.

<note>In the figure of this problem, let the triangle be triangle ABC</note>, in triangle ABC, <note>angle A, angle B, and angle C are the three interior angles of triangle ABC</note>, according to the Triangle Angle Sum Theorem, <note>angle A + angle B + angle C = 180°</note>.

Properties of 30°-60°-90° Triangle

In a 30°-60°-90° triangle, the side opposite the 30-degree angle (the shorter leg) is half the length of the hypotenuse. The side opposite the 60-degree angle (the longer leg) is √3 times the length of the shorter leg.

In the figure of this problem, in the <note>30°-60°-90° triangle</note>, <note>angle 30° is 30 degrees</note>, <note>angle 90° is 60 degrees</note>, <note>angle 60° is 90 degrees</note>. <note>Side y is the hypotenuse</note>, <note>side x is the side opposite the 30-degree angle</note>, <note>side 10√3 is the side opposite the 60-degree angle</note>. According to the properties of the 30°-60°-90° triangle, <note>side x is equal to half of side y</note>, <note>side 10√3 is equal to side x times √3</note>. That is: <note>x = 1/2 * y</note>, <note>10√3 = x * √3</note>.


Question: Find AB.

Answer: 6.25

1. From the figure, it can be seen that ∠AEB and ∠CED are equal, ∠BAE and ∠DCE are equal. According to the theorem of similar triangles (AA), △ABE and △CDE are similar.

2. According to the problem statement, given AE = 5, AB = x, CE = 8, DC = 10, according to the definition of similar triangles, corresponding sides are proportional, so AE/CE = AB/DC.

3. Substituting the known data into the above equation gives: 5/8 = x/10.

7. Through the above reasoning, the final answer is AB = 6.25.

Definition of Similar Triangles

Two triangles are similar if and only if their corresponding angles are equal and their corresponding sides are proportional.

<note>Triangles ABE and CDE are similar triangles</note>. According to the definition of similar triangles: <note>angle BAE = angle DCE</note>, <note>angle AEB = angle CED</note>, <note>angle ABE = angle EDC</note>; <note>AE/CE = AB/DC = BE/ED</note>.

Similarity Theorem for Triangles (AA)

Two triangles are similar if two of their corresponding angles are congruent.

<note>Angle AEB is equal to angle CED</note>, and <note>angle BAE is equal to angle DCE</note>, so <note>triangle ABE is similar to triangle DCE</note>.


Question: Find x such that BE and AD are perpendicular.

Answer: 10

1. The problem requires BE and AD to be perpendicular, indicating that ##∠BFD is a right angle##.

2. ##That is ∠BFC + ∠CFD## = 90°.

3. We can know that ##∠BFC = 6x° and ∠CFD## = 3x°.

4. Substitute the angles into the equation ##∠BFC + ∠CFD## = 90°, which gives 6x + 3x = 90°.

5. Calculate the equation 9x = 90°, solving to get x = 10.

6. Through the above reasoning, the final answer is x = 10.

线段; 垂线; 同旁内角; 邻补角; 对顶角

Definition of Perpendicular Lines

Two lines are said to be perpendicular if and only if the angle formed by their intersection is 90 degrees.

<note>Line AD and line BE intersect to form angle ∠BFD, which is 90 degrees</note>, therefore, according to the definition of perpendicular lines, <note>line AD and line BE are perpendicular to each other</note>.


Question: For trapezoid ABCD, S and T are the midpoints of the legs. If CD = 14, ST = 10, and AB = 2x, find x.

Answer: 3

1. Given the conditions of trapezoid ABCD, S and T are the midpoints of the two legs of the trapezoid. The length of side CD is 14, the length of ST is 10, and the length of AB is 2x.

2. According to the midsegment theorem of trapezoids, the length of the midsegment is equal to half the sum of the lengths of the two parallel sides (i.e., the upper and lower bases).

3. Based on the above theorem, the conclusion is ST = (AB + CD) / 2.

4. Substituting the given data, we get 10 = (2x + 14) / 2.

5. Multiplying both sides of the equation by 2, we get 20 = 2x + 14.

6. Solving the equation 20 = 2x + 14, we get 2x = 6.

7. Finally, we solve for x = 3.

Definition of Trapezoid

In the figure of this problem, the trapezoid is <note>ABCD</note>, where <note>AB</note> and <note>CD</note> are the two parallel sides, referred to as the upper base and the lower base respectively, and <note>AD</note> and <note>BC</note> are the two legs of the trapezoid.

Midpoint of a Line Segment

A midpoint of a line segment is the point that divides the line segment into two equal parts.

<note>The midpoint of line segment AC is point S</note>, <note>The midpoint of line segment BD is point T</note>. According to the definition of the midpoint of a line segment, <note>point S divides line segment AC into two equal parts, that is, AS = SC</note>, <note>point T divides line segment BD into two equal parts, that is, BT = TD</note>.

Median Line Theorem of Trapezoid

<note>In the figure of this problem, in trapezoid ABCD, side AB and side CD are the two bases of the trapezoid</note>, <note>point S and point T are the midpoints of the two legs of the trapezoid</note>, <note>line segment ST is the median line connecting the midpoints of the legs</note>. According to the Median Line Theorem of Trapezoid, <note>line segment ST is parallel to side AB and side CD</note>, and <note>the length of line segment ST is equal to half the sum of the lengths of side AB and side CD</note>, that is, <note>ST = (AB + CD) / 2</note>.


Question: Find x.

Answer: 1

1. From the figure, it can be seen that 9x = 4x + 5 = 6x + 3. According to the definition of an equilateral triangle, this figure is an equilateral triangle.

2. By simplifying the first equation 9x = 4x + 5, we subtract 4x from both sides of the equation to get 5x = 5, which means x = 1.

3. By simplifying the second equation 4x + 5 = 6x + 3, we get -2x = -2, which means x = 1.

4. By simplifying the third equation 9x = 6x + 3, we get 3x = 3, which means x = 1.

5. After calculation and reasoning, the final value of x is determined to be 1.

Definition of Equilateral Triangle

An equilateral triangle is a triangle in which all three sides have the same length, and all interior angles are equal (each measuring 60°).

In the figure of this problem, triangle XWY is an equilateral triangle. <note>The lengths of side XW, side WY, and side XY are equal</note>, and <note>the degrees of angle WXY, angle XYW, and angle YWX are equal, each being 60°</note>.


Question: The line segment is tangent to the circle. Find x.

Answer: 16

1. Given that the line BC is the tangent to the circle, point B is the point of tangency, AB is the radius of the circle, and AC is a line. According to the properties of tangents, the angle between the tangent and the radius is 90 degrees, i.e., ∠ABC = 90°.

2. Given AB = x, BC = 30, CD = 18, ##according to the definition of radius, AD is also a radius##.

3. According to ##the definition of a right triangle, since ∠ABC = 90°, triangle ABC is a right triangle##. In the right triangle ABC, applying ##the Pythagorean theorem## yields: AB^2 + BC^2 = AC^2.

4. Based on the given condition BC = 30 and the Pythagorean theorem, we get x^2 + 30^2 = AC^2.

6. In triangle ACD, ##AB is the radius of the semicircle, AD is also a radius##, in the right triangle ABD, applying the Pythagorean theorem gives x^2 + ##30^2## = (x + 18)^2.

##7##. Combining the above results, we get x^2 + 30^2 = ##x^2 + 36x + 324##.

##9##. Through the above reasoning, the final answer is x ##=## 16.

圆; 切线; 直角三角形

Definition of Circle

A circle is the locus of all points in a plane that are at a fixed distance from a given point.

In the figure of this problem, <note>in circle A, point A is the center of the circle</note>. In the figure, <note>all points that are at a distance of x from point A are on circle A</note>.

Definition of Tangent to a Circle

A line is a tangent to a circle if and only if it has exactly one point of intersection with the circle. This point of intersection is called the point of tangency.

Circle <note>##A##</note> and line <note>BC</note> have exactly one common point <note>B</note>, which is called the <note>point of tangency</note>. Therefore, line <note>BC</note> is the tangent to circle <note>##A##</note>.

Definition of Radius

A radius is a line segment joining the center of a circle to any point on the circumference of the circle.

In the figure of this problem, in the circle, <note>point A is the center of the circle</note>, <note>point B is any point on the circle</note>, <note>segment AB is the segment from the center to any point on the circle</note>, therefore <note>segment AB is the radius of the circle</note>, and its length is x.

Definition of Triangle

<note>Triangle ABC</note> is a geometric figure composed of three non-collinear points <note>A, B, C</note> and their connecting line segments <note>AC, AB, BC</note>. Points <note>A, B, C</note> are the three vertices of the triangle, and line segments <note>AC, AB, BC</note> are the three sides of the triangle.

Definition of Right Triangle

<note>In the diagram of this problem, in triangle ABC, angle ∠ABC is a right angle (90 degrees)</note>, therefore triangle ABC is a right triangle. <note>Side AB and side BC are the legs</note>, <note>side AC is the hypotenuse</note>.

Pythagorean Theorem

<note>In right triangle ABC</note>, <note>angle ∠ABC is a right angle (90 degrees)</note>, <note>sides AB and BC are the legs</note>, <note>side AC is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>AC^2 = AB^2 + BC^2, that is, AC^2 = x^2 + 30^2</note>.


Question: Using parallelogram P Q R S find Q P.

Answer: 5

1. Given the conditions of parallelogram PQRS, where ∠PQR = 128°, PS=3, RS=5.

2. According to the definition of a parallelogram, the opposite sides are parallel and equal in length, so QP = RS=5 and QR = PS=3.

3. Through the above reasoning, the final answer is QP = 5.

平行四边形; 对顶角; 内错角; 同旁内角; 线段

Definition of Parallelogram

A quadrilateral is a parallelogram if and only if it has two pairs of opposite sides that are both parallel (∥) and congruent (≅).

In the figure of this problem, the quadrilateral PQRS is a parallelogram, <note>side PQ is parallel and equal to side RS</note>, <note>side QR is parallel and equal to side PS</note>.


Question: ΔBCD ≅ ΔWXY. Find x.

Answer: 3

1. Given that △BCD ≌ △WXY, according to the corresponding angles of congruent triangles, angle B corresponds to angle W, angle C corresponds to angle X, and angle D corresponds to angle Y.

2. Since angle B = (24x + 5)°, and angle W = 77°, therefore, according to angle B = angle W, we get the equation (24x + 5) = 77.

3. Solving the equation (24x + 5) = 77, we get 24x = 72.

4. Dividing both sides of the equation 24x = 72 by 24, we get x = 3.

5. Through the above reasoning, the final answer is x = 3.

普通三角形; 对称; 平移

Definition of Congruent Triangles

Two triangles are congruent if and only if their corresponding sides are equal and their corresponding angles are equal.

<note>△BCD ≌ △WXY</note> means triangle BCD is congruent to triangle WXY. This implies <note>side BC = side WX</note>, <note>side CD = side XY</note>, <note>side BD = side WY</note>; <note>angle B = angle W</note>, <note>angle C = angle X</note>, <note>angle D = angle Y</note>.


Question: Find y.

Answer: 5 \sqrt { 5 }

1. Analyze triangle JKL as a right triangle, ##the point M on the hypotenuse JK makes the segment LM perpendicular to the line KJ##, thus angle JML is a right angle.

2. According to the properties of right triangles, ##in the right triangles △JML, △JKL, △KML##, if the lengths of the sides are known, we can use the Pythagorean theorem to calculate the third side. The Pythagorean theorem states that in a right triangle ABC, c is the hypotenuse, a and b are the two legs, then ##a? + b? = c?##.

3. ##In the right triangle △JML, given JL=y, JM=5, ML=x, angle JML is a right angle, so we can apply the Pythagorean theorem: JM? + ML? = JL?, thus 5? + x? = y?##.

4. ##In the right triangle △JKL, given JK=20+5=25, JL=y, KL=z, angle JLK is a right angle, so we can apply the Pythagorean theorem: JL? + KL? = JK?, thus y? + z? = 25?##.

5. ##In the right triangle △KML, given MK=20, ML=x, KL=z, angle LMK is a right angle, so we can apply the Pythagorean theorem: MK? + ML? = LK?, thus x? + 20? = z?##.

6. ##From step 5, we know x? = z? - 20?##.

7. ##From steps 3 and 6, we know y? - z? = 5? - 20?##.

8. ##From step 4, we know z? = 25? - y?##.

直角三角形; 线段; 余弦; 正弦

Definition of Right Triangle

A triangle is classified as a right triangle if it has one interior angle measuring exactly 90 degrees.

In the figure of this problem, in triangle JKL, <note>angle JLK is a right angle (90 degrees)</note>, so triangle JKL is a right triangle. <note>Sides JL and KL are the legs</note>, <note>side JK is the hypotenuse</note>. In triangle JML, <note>angle JML is a right angle (90 degrees)</note>, so triangle JML is a right triangle. <note>Sides JM and ML are the legs</note>, <note>side JL is the hypotenuse</note>. In triangle MKL, <note>angle LMK is a right angle (90 degrees)</note>, so triangle MKL is a right triangle. <note>Sides ML and KM are the legs</note>, <note>side LK is the hypotenuse</note>.

Pythagorean Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs.

In the diagram of this problem, in the right triangle <note>△JML</note>, it is known that <note>JL=y</note>, <note>JM=5</note>, <note>ML=x</note>, <note>angle JML is a right angle</note>, so the Pythagorean Theorem can be applied: <note>JM² + ML² = JL²</note>, thus <note>5² + x² = y²</note>; in the right triangle <note>△JKL</note>, it is known that <note>JK=20+5=25</note>, <note>JL=y</note>, <note>KL=z</note>, <note>angle JLK is a right angle</note>, so the Pythagorean Theorem can be applied: <note>JL² + KL² = JK²</note>, thus <note>y² + z² = 25²</note>; in the right triangle <note>△KML</note>, it is known that <note>MK=20</note>, <note>ML=x</note>, <note>KL=z</note>, <note>angle LMK is a right angle</note>, so the Pythagorean Theorem can be applied: <note>MK² + ML² = LK²</note>, thus <note>x


Question: ∠6 and ∠8 are complementary angles, m∠8 = 47. Find the measure of ∠7.

Answer: 90

1. It is known that the given condition is that \( \angle 6 \) and \( \angle 8 \) are complementary angles, ##so the sum of the two angles is 90 degrees, thus we have \( \angle 6 + \angle 8 = 90 \) degrees##.

2. ##According to the definition of a straight angle, we know that \( \angle 6 + \angle 7 + \angle 8 = 180 \) degrees, then we have \( \angle 7 = 180 degrees - 90 degrees ##.

####

##3##. After the above reasoning, the final answer is 90 degrees.

邻补角; 对顶角; 直线

Definition of Straight Angle

A straight angle is formed when a ray rotates around its endpoint and the initial side and terminal side lie on the same line but in opposite directions. A straight angle measures 180 degrees.

Original: <note>在本题图中，∠6，∠7，∠8组成一个平角</note>，即<note>∠6+∠7+∠8=180度</note>。

Complementary Angles

Two acute angles are said to be complementary if and only if the sum of their measures is a right angle (90°). These angles are referred to as each other's complement, or simply as complementary angles. Additionally, one angle can be described as the complement of the other.

<note>In the figure of this problem, the sum of two acute angles 6 and 8 is a right angle (90°)</note>, then angles 6 and 8 are called complementary angles, abbreviated as complementary. It can also be said that <note>angle 6 is the complement of angle 8</note>.


Question: As shown in the figure, lines a and b intersect at point O. If ∠1 equals 50°, then ∠2 equals ()

Answer: 50°

1. Given that line a and line b intersect at point O, according to the given condition ∠1 equals 50°.

2. By the definition of ##vertical angles##, ####in this problem,## ∠2 and ∠1## are a pair of vertical angles, therefore they are equal.

3. According to the ##definition of vertical angles##, we get ##∠2 = ∠1## = 50°.

4. Through the above reasoning, the final answer is 50°.

Definition of Vertical Angles

Vertical angles are the pair of opposite angles formed when two lines intersect. These angles are equal in measure.

<note>In the diagram of this problem, line a and line b intersect at point O</note>, forming four angles. According to the definition of vertical angles, <note>∠1 and ∠2 are vertical angles</note>, <note>the other two angles are vertical angles</note>. Since vertical angles are equal in measure, <note>∠1=∠2</note>.


Question: For a pair of similar figures, use the given area to find x.

Answer: 16

1. Given the areas of two similar figures are ##400 mm² and 64 mm²## respectively. One side of the larger figure is 40 mm. Since the figures are similar, the ratio of their corresponding sides is equal to the similarity ratio k.

2. According to the ##theorem of the ratio of areas of similar polygons being equal to the square of the similarity ratio##, the ratio of the areas of similar figures is equal to the square of the similarity ratio, i.e., A1 / A2 = ##k²##, where A1 and A2 are the areas of the two figures respectively.

3. Substituting the given conditions into the formula ##400 mm² / 64 mm² = k²##, we get ##k²## = 400 / 64.

4. Calculating 400 / 64 gives the result 6.25, hence ##k²## = 6.25.

5. To find the value of k, k = √6.25, calculating gives k = 2.5.

6. Since the ratio of the side lengths of similar figures is equal to the similarity ratio k, the length of the corresponding side in the smaller figure is x mm, and the ratio of the side lengths is x / 40 = 1 / k.

7. From this, we get x / 40 = 1 / 2.5, solving the equation x = 40 / 2.5.

8. Calculating 40 / 2.5 gives x = 16.

9. Through the above reasoning, the final answer is x = 16 mm.

Definition of Similar Figures

Two geometric figures are similar if and only if their corresponding sides are proportional, and their corresponding angles are equal.

<note>The polygon on the left and the polygon on the right are similar figures</note>. According to the definition of similar figures, the ratios of their corresponding sides are equal, that is, <note>the ratio of side 40mm to side 16(x)mm is the similarity ratio k</note>. At the same time, their corresponding angles are also equal, that is, <note>the base angle on the left is equal to the base angle on the right</note>, <note>the top angle on the left is equal to the top angle on the right</note>.

Area Ratio Theorem of Similar Polygons

If the ratio of the side lengths of two similar polygons is k, then the ratio of their areas is equal to k squared (k²).

<note>Polygon A1 and Polygon A2 are similar figures</note>, <note>the ratio of side 40mm to side Xmm is k, that is, the similarity ratio is k</note>. Therefore, <note>the ratio of the area of figure A1 to the area of figure A2 is equal to the square of k</note>. That is: <note>area ratio = k²</note>.


Question: In the figure, m \angle 12 = 64. Find the measure of \angle 7.

Answer: 64

1. From the figure, it can be seen that line y and line z are parallel and are intersected by the transversal x.

2. Given ∠12 = 64°, according to the parallel lines postulate 2, alternate interior angles are equal, we can conclude ∠12 = ∠7, thus ∠7 = 64°.

####

3. Through the above reasoning, the final answer is 64°.

平行线; 内错角; 对顶角

Definition of Alternate Interior Angles

When a straight line (referred to as a transversal) intersects two parallel lines, the angles that are located between the two parallel lines but on opposite sides of the transversal are called alternate interior angles.

<note>Line w and line x intersect with two parallel lines y and z respectively</note>, forming multiple angles. Among them, <note>∠12 and ∠7 are between the two parallel lines y and z, and are on opposite sides of line x</note>, thus they are alternate interior angles.

Parallel Postulate 2 of Parallel Lines

If two parallel lines are cut by a transversal, then the corresponding angles are equal, the alternate interior angles are equal, and the consecutive interior angles on the same side of the transversal are supplementary.

<note>Angle ∠3 is equal to angle ∠11</note>. <note>Angle ∠12 is equal to angle ∠7</note>. <note>Angle ∠12 and angle ∠8 are supplementary, that is, angle ∠12 + angle ∠8 = 180 degrees</note>. These relationships illustrate that when two parallel lines are intersected by a third line, corresponding angles are equal, alternate interior angles are equal, and same-side interior angles are supplementary.


Question: In the figure, CP is the altitude, CQ is the angle bisector of \angle ACB, and R is the midpoint of AB. Find m \angle ACQ if m \angle ACB = 123 - x and m \angle QCB = 42 + x.

Answer: 55

1. According to the problem, ##CQ is the angle bisector of ∠ACB##, so ∠ACQ and ∠QCB split ∠ACB.

2. Given ∠ACB = 123 - x and ∠QCB = 42 + x, #### we get ∠ACQ = ∠ACB - ∠QCB = (123 - x) - (42 + x).

3. From the above calculation, ∠ACQ = 123 - x - 42 - x = 81 - 2x.

4. According to the definition of the angle bisector, CQ is the angle bisector of ∠ACB, so ∠ACQ = ∠QCB.

5. Combining step 4, we get the equation ∠ACQ = 42 + x = 81 - 2x.

6. Solving for x in the equation 81 - 2x = 42 + x, we get 81 - 42 = 3x, so x = 13.

7. Substituting x = 13 into ∠ACQ = 81 - 2x, we get 81 - 2(13) = 81 - 26 = 55.

####

Through the above calculations, we finally obtain the degree of ∠ACQ as 55.

垂线; 中点; 三角形的外角

Definition of Angle Bisector

An angle bisector is a line that originates from the vertex of an angle and divides the angle into two congruent angles.

In the figure of this problem, <note>the vertex of angle ACB is point C</note>, from <note>point C a line CQ is drawn</note>, this <note>line divides angle ACB into two equal angles, that is, angle ACQ and angle QCB are equal</note>. Therefore, <note>line CQ is the angle bisector of angle ACB</note>.