Question: As shown in the figure, in the right triangle △ABC, ∠C=90°, AC=3, BC=4, find the value of \cosA.

Answer: \frac{3}{5}

1. Given in the right triangle ABC, ∠C = 90°, AC = 3, BC = 4, based on the given conditions we need to calculate side AB.

2. ##According to the Pythagorean theorem##, in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse, that is: AB^2 = AC^2 + BC^2.

3. ##Substitute the given conditions##, that is: AB = √(AC^2 + BC^2) = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.

4. Thus, the length of the hypotenuse AB is 5.

5. ##According to the cosine function, the cosine of ∠A is equal to the length of the adjacent side divided by the length of the hypotenuse.##

6. ##Specifically in this problem, cos(∠A) = adjacent side AC / hypotenuse AB = 3 / 5.##

7. Through the above reasoning, the final answer is 3/5.

直角三角形; 余弦; 线段

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.

<note>In the figure of this problem, in the right triangle ABC, side AC is the adjacent side of ∠A, and side AB is the hypotenuse.</note> Therefore, according to the definition of the cosine function, <note>the cosine value of ∠A is equal to the ratio of the adjacent side AC to the hypotenuse AB, that is, cos(∠A) = AC / AB = 3 / 5.</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.

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


Question: Fold a rectangular strip of paper with equal width as shown in the figure. If ∠1 = 140°, then the degree of ∠2 is ()

Answer: 110°

1. Let the alternate interior angle of ∠1 formed by folding be ∠BAD = ∠CAB + ∠ACD. It is known that ∠1 = 140°.

2. According to the parallel postulate 2 of parallel lines, we have ∠CAB + ∠ACD = ∠1 = 140°.

3. According to the property of the folded figure, when folded once, ∠CAB = ∠ACD.

4. Let ∠CAB = ∠ACD = x.

5. From steps 2 and 3, we get 2x = 140°.

6. Solving this, we get x = 70°, i.e., ∠CAB = ∠ACD = 70°.

7. According to the supplementary angles on the same side, we get ∠2 = 180° - ∠ACD = 180° - 70° = 110°.

8. Through the above reasoning, the final answer is ∠2 = 110°.

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.

In the figure of this problem, two parallel lines (the upper and lower edges of the paper) are intersected by a line, where angle ∠1 and ∠BAD are located between the two parallel lines and on opposite sides of the intersecting line (the crease). Therefore, angle ∠1 and ∠BAD are alternate interior angles. Alternate interior angles are equal, that is, ∠1 is equal to ∠BAD.

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>The top and bottom edges of the paper form a parallel relationship, and the crease is a transversal</note>, so <note>alternate interior angles ∠1 = ∠BAD</note>, <note>same-side interior angles ∠2 + ∠BAD = 180°</note>.

Property of Angle Bisector

Any point on the angle bisector is equidistant to the two sides of the angle.

<note>∠BAD</note> is divided into two equal angles, namely <note>∠CAB and ∠ACD</note>. According to the property of the angle bisector, let these two angles be x each, then 2x = ∠1 = 140°, so x = 70°, meaning each of these two angles is 70°.

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.

<note>In the figure of this problem, two parallel lines (the upper and lower edges of the strip of paper)</note> are intersected by <note>a line (the crease)</note>. The angles <note>∠2 and ∠BAD</note> are on the same side of the transversal and inside the two intersected lines, so <note>∠2 and ∠BAD are consecutive interior angles</note>. Consecutive interior angles <note>∠2 and ∠BAD are supplementary</note>, that is, <note>∠2 + ∠BAD = 180 degrees</note>.


Question: As shown in the figure, the diagram is a kite frame made by Xiao Liu. Given BC∥PQ, AB:AP=2:5, AQ=20cm, find the length of CQ.

Answer: 12cm

1. Given BC∥PQ, according to the parallel postulate 2 of parallel lines and the definition of corresponding angles, ∠ABC=∠APQ, and at the same time ∠BAC=∠PAQ. According to the AA criterion for similar triangles, △ABC is similar to △APQ.

2. Since △ABC is similar to △APQ, according to the proportional property of similar triangles, there is a proportional relationship \\(rac{AB}{AP}=rac{AC}{AQ}\\).

3. According to the given conditions AB:AP=2:5, AQ=20cm, we can substitute into the proportional relationship \\(rac{AC}{20} = rac{2}{5}\\).

4. By calculating the proportion, we find AC=8cm.

5. Therefore, the length of CQ is AQ - AC.

6. Calculating, we get CQ = 20cm - 8cm = 12cm.

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

平行线; 普通三角形; 位似; 线段

Definition of Parallel Lines

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

The line segments <note>BC and PQ</note> lie in the same plane and do not intersect, so according to the definition of parallel lines, <note>the line segments BC and PQ</note> are parallel lines.

Definition of Similar Triangles

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

<note>△ABC与△APQ相似</note>, that is, <note>corresponding angles are equal</note>, <note>corresponding sides are proportional</note>. For example, <note>∠BAC = ∠PAQ</note>, <note>∠ABC = ∠APQ</note>, and <note>∠ACB = ∠AQP</note>, and <note>\(\frac{AB}{AP}=\frac{BC}{PQ}=\frac{AC}{AQ}\)</note>.

Parallel Postulate 2 of Parallel Lines

<note>Two parallel lines BC and PQ</note> are intersected by a third line AP, forming the following geometric relationships: 1. Corresponding angles: <note>angle ABC and angle APQ are equal</note>. These relationships indicate 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 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 BC and PQ</note> are intersected by <note>a transversal AP</note>, where <note>angle ABC and angle APQ are on the same side of the transversal AP and on the same side of the intersected lines BC and PQ</note>, therefore <note>angle ABC and angle APQ are corresponding angles</note>. Corresponding angles are equal, that is <note>angle ABC is equal to angle APQ</note>.

Similarity Theorem for Triangles (AA)

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

<note>Angle ABC is equal to angle APQ</note>, and <note>angle BAC is equal to angle PAQ</note>, so <note>triangle ABC is similar to triangle APQ</note>.


Question: As shown in the figure, in △ABC, ∠C=90°, point D is a moving point on side BC (point D does not coincide with point C). The circle with CD as its diameter intersects AD at point P. If AC=6, the minimum length of segment BP is 2. Then the length of AB is ()

Answer: 2√{13}

1. Given CD is the diameter, according to the ##(corollary 2 of the inscribed angle theorem) the inscribed angle subtended by the diameter is a right angle##, we get ∠CPD=90°.

2. ##Because the circle with CD as the diameter intersects AD at point P, according to the definition of a straight angle, ∠CPD+∠APC=180°, so ∠APC=180°-∠CPD=180°-90°=90°, by ##(corollary 2 of the inscribed angle theorem) the inscribed angle subtended by the diameter is a right angle##, we conclude that point P is on the circle with AC as the diameter (denoted as ⊙O).

3. ##According to the problem condition, the minimum length of segment BP is 2. When points B, P, and O are collinear, BP reaches its minimum value. Let BP' = 2, then OB=OP'+BP'##.

4. ##For ⊙O with AC as the diameter##, AC = 6, hence ##radius OA = OC = 3##.

5. ##So according to the definition of radius, OP'=3, substituting into the equation gives OB=OP'+BP'=3+2=5##.

6. ##Since ∠C=90°, according to the definition of a right triangle, triangles ABC and OBC are right triangles. In right triangle OBC, OC = 3, OB = 5, according to the Pythagorean theorem, we get OB²=BC²+OC², substituting the values we get, 5²=BC²+3² => BC²=5²-3²=16 => BC = √16 = 4##.

7. In right triangle ##ABC##, ∠C = 90°, AC = 6, BC = 4, ##according to the Pythagorean theorem, we get AB²=BC²+AC², substituting the values we get, AB²=4²+6²=52 => AB = √52 = 2√13##.

8. Through the above reasoning, we finally get the length of AB as 2√13.

直角三角形; 圆; 圆周角

Definition of Radius

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

<note>In the figure of this problem</note>, in the circle O with AC as the diameter, <note>point O is the center of the circle</note>, points A, C, P, P' are any points on the circle, the line segments <note>OA, OC, OP'</note> are segments from the center of the circle to any point on the circle, therefore the line segments <note>OA, OC, OP'</note> are the radii of the circle.

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 ABC, <note>angle C is a right angle (90 degrees)</note>, therefore <note>triangle ABC is a right triangle</note>. <note>Side BC and side AC are the legs</note>, <note>side AB is the hypotenuse</note>. Similarly, in triangle OBC, <note>angle C is a right angle (90 degrees)</note>, therefore <note>triangle OBC is a right triangle</note>. <note>Side BC and side OC are the legs</note>, <note>side BO is the hypotenuse</note>.

Corollary to the Inscribed Angle Theorem 2: The Angle Subtended by the Diameter

The angle subtended by the diameter of a semicircle is a right angle; conversely, if an inscribed angle is 90°, then the chord subtending that angle is a diameter.

<note>In a circle with CD as the diameter, the angle subtended by the diameter CD at the circumference, ∠CPD, is a right angle (90 degrees)</note>, because <note>∠APC=90°</note>, so <note>point P is on the circle with AC as the diameter (denoted as ⊙O)</note>.

Pythagorean Theorem

<note>In the right triangle ABC, angle C is a right angle (90 degrees)</note>, <note>side BC and side AC are the legs</note>, <note>side AB is the hypotenuse</note>, so according to the Pythagorean Theorem, AB² = BC² + AC². <note>In the right triangle OBC, angle C is a right angle (90 degrees)</note>, <note>side BC and side OC are the legs</note>, <note>side BO is the hypotenuse</note>, so according to the Pythagorean Theorem, OB² = BC² + OC².

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.

<note>AD intersects the circle with CD as its diameter at point P</note>, then <note>A, P, D are on the same line</note>, so <note>∠CPD + ∠APC = 180°</note>.


Question: As shown in the figure, the light source P is directly above the horizontal bar AB. The shadow of AB under the light is CD, AB∥CD, AB=2m, CD=5m, the distance from point P to CD is 3m, then the distance between AB and CD is ()m.

Answer: \frac{9}{5}

1. Draw a perpendicular from point P to CD, intersecting lines AB and CD at points E and F respectively. According to the definition of perpendicular, we can deduce that PE ⊥ AB and PF ⊥ CD. Thus, ∠PEA = ∠PFC = 90 degrees. According to the definition of right triangles, triangles PEA and PFC are right triangles.

2. From the figure, we know that AB is parallel to CD. According to the parallel axiom 2 of parallel lines, corresponding angles are equal, so ∠PAE = ∠PCF.

3. According to the similarity of right triangles and the definition of similar triangles, △PEA and △PFC are similar, and their corresponding sides are proportional, i.e., PE / PF = AE / CF.

4. Given AB = 2 meters and CD = 5 meters, according to the definition of perpendicular bisector, AE = 2 / AB = 1 meter and CF = 2 / CD = 2.5 meters. The distance from point P to CD, PF, is 3 meters. Let the distance from point P to AB be PE = x meters. Substituting the data into the above proportion equation, we get x / 3 = 1 / 2.5.

5. Solving the proportion equation, we get x = (1 / 2.5) * 3 = 6 / 5 meters.

6. Solving for x = 6 / 5, we find that the distance from point P to AB, PE, is 6 / 5 meters.

7. From the figure, we can deduce that the distance between AB and CD is PF - PE = 3 - 6 / 5 meters.

8. After calculation, the distance between AB and CD is (15 / 5) - (6 / 5) = 9 / 5 meters.

9. Finally, we find that the distance between AB and CD is 1.8 meters.

平行线; 等腰三角形; 直角三角形

Similarity of Right Triangles

Two right triangles are similar if and only if one of their acute angles are equal.

<note>In right triangles PEA and PFC, angles PEA and PFC are right angles (90 degrees)</note>, and <note>angle PEA is equal to angle PFC</note>, therefore, according to the similarity theorem of right triangles, these two right triangles are similar. That is, <note>the corresponding sides of triangles PEA and PFC are proportional</note>.

Definition of Similar Triangles

In the figure of this problem, <note>triangle PAB and triangle PCD are similar triangles</note>. According to the definition of similar triangles: <note>∠PEA = ∠PFC</note>, <note>∠PAE = ∠PCF</note>, <note>∠APE = ∠CPF</note>; <note>PE / PF = AE / CF</note>. Given that AB = 2 meters, CD = 5 meters, <note>AE = 2/AB=1 meter</note>, <note>CF = 2/CD=2.5 meters</note>, let the distance from point P to AB be x meters, and the distance from point P to CD be 3 meters. According to the proportional relationship of corresponding sides of similar triangles: <note>x/3 = 1/2.5</note>. By calculation, x = <note>(1/2.5)</note> * 3 = 6/5 meters, thus determining the distance between AB and CD.

Parallel Postulate 2 of Parallel Lines

In the figure of this problem, <note>two parallel lines AB and CD are intersected by a third line PC</note>, forming the following geometric relationship: <note>corresponding angles: angle PAE and angle PCF are equal</note>.

Definition of Corresponding Angles

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

Definition of Altitude

An altitude is a perpendicular line segment extending from a vertex of a triangle to the opposite side (or to the extension of the opposite side).

<note>The line segment PF perpendicular to the opposite side CD (or its extension) from vertex P is the altitude of vertex P</note>. <note>The line segment PF forms a right angle (90 degrees) with side CD (or its extension)</note>, which indicates that the line segment PF is the perpendicular distance from vertex P to the opposite side CD (or its extension).

Definition of Perpendicular Bisector

A line is a perpendicular bisector if and only if it passes through the midpoint of a line segment and is perpendicular to that line segment.

In the figure of this problem, <note>line PE passes through the midpoint of segment AB</note>, and <note>line PE is perpendicular to segment AB</note>. Therefore, line PE is the perpendicular bisector of segment AB; <note>line PF passes through the midpoint of segment CD</note>, and <note>line PF is perpendicular to segment CD</note>. Therefore, line PF is the perpendicular bisector of segment CD.


Question: Find x.

Answer: 4 \sqrt { 2 }

1. Let the three vertices of the triangle be A, B, C. Given ∠ABC=90°, according to the definition of a right triangle, △ABC is a right triangle. According to the triangle angle sum theorem, given ∠BAC=45°, ∠ACB=180°-∠ABC-∠CAB=180°-90°-45°=45°.

2. In the right triangle ABC, since ∠BAC=∠ACB=45°, this indicates that this triangle is a special right triangle with angles 45°-45°-90°, which is an isosceles right triangle.

3. According to the properties of a 45°-45°-90° triangle, the lengths of the two legs are equal, and the hypotenuse is √2 times the length of a leg.

4. In this triangle, the length of BC is 4. According to the definition of an isosceles right triangle, the length of AB is also 4.

5. The length of the hypotenuse AC, according to the properties of a 45°-45°-90° triangle, is AC=BC*√2=4 × √2, which means x=4√2.

6. Through the above reasoning, the final answer is 4√2.

直角三角形; 正弦; 余弦

Definition of Right Triangle

In the figure of this problem, <note>one angle in the triangle is a right angle (90 degrees)</note>, so this triangle is a right triangle. It is known that <note>the length of one of the legs is 4</note>.

Definition of Isosceles Right Triangle

An isosceles right triangle is a triangle with two sides of equal length and one angle measuring 90 degrees.

<note>In the figure of this problem, triangle ABC is an isosceles right triangle</note>, in which <note>angle ABC is a right angle (90 degrees)</note>, <note>side AB and side BC are equal right-angle sides</note>.

Triangle Angle Sum Theorem

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

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

Properties of 45°-45°-90° Triangle

In a 45°-45°-90° triangle, the two legs opposite the 45-degree angles are congruent, and the hypotenuse is √2 times the length of either leg.

<note>In the diagram of this problem, in the 45°-45°-90° triangle ABC, angle BAC is 45 degrees, angle BCA is 45 degrees, and angle ABC is 90 degrees</note>.<note>Side AC is the hypotenuse, sides AB and BC are the legs opposite the two 45-degree angles</note>. According to the properties of the 45°-45°-90° triangle, <note>side AB is equal to the length of side BC, and side AC is √2 times the length of side BC</note>. That is: <note>AB = BC, AC = BC * √2</note>.


Question: Refer to the figure on the right. If p \parallel q, find the value of m \angle F C E.

Answer: 105

1. Given that line p is parallel to line q, according to the parallel lines axiom 2, we have ∠DBF = ∠ECF (corresponding angles are equal).

2. The degree of ∠ECF is (2x + 15)°, and ∠DBF is y°.

3. Thus, we can set (2x + 15) = y. Also, according to the definition of adjacent supplementary angles and the definition of a straight angle, we know that ∠ABD and ∠DBF are supplementary, so (3x - 60) + y = 180°.

4. From this, we obtain two equations: (2x + 15) = y and (3x - 60) + y = 180°.

5. Solving the system of equations, first we get y = (2x + 15) from the first equation.

6. Substituting y = (2x + 15) into the second equation, we get (3x - 60) + (2x + 15) = 180°.

7. Combining like terms, we get 5x - 45 = 180°.

8. Moving 45 to the right side, we get 5x = 225.

9. Dividing both sides by 5, we get x = 45.

10. Substituting x = 45 into y = (2x + 15), we get y = 2(45) + 15 = 105°.

11. Finally, we find that ∠FCE = ∠DBF = y = 105°.

12. After the detailed reasoning above, the final answer is that the degree of ∠FCE is 105°.

平行线; 内错角; 邻补角

Definition of Corresponding Angles

<note>Two parallel lines p and q are intersected by a line AF</note>, where <note>angle DBF and angle ECF are on the same side of the intersecting line AF and on the same side of the intersected lines p and q</note>, therefore <note>angle DBF and angle ECF are corresponding angles</note>. Corresponding angles are equal, that is <note>angle DBF is equal to angle ECF</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.

Original: <note>Two parallel lines p and q</note> are intersected by <note>a third line AF</note>, forming the following geometric relationships: <note>Corresponding angles: angle ABD and angle BCE are equal, angle FCE and angle CBD are equal</note>. These relationships indicate that when two parallel lines are intersected by a third line, the corresponding angles are equal, the alternate interior angles are equal, and the same-side interior angles are supplementary.

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.

In the figure of this problem, <note>angle ABD and angle FBD share a common side BD</note>, <note>their other sides AB and FB are extensions of each other in opposite directions</note>, so <note>angle ABD and angle FBD are adjacent supplementary angles</note>.

Definition of Straight Angle

<note>Ray AF rotates around endpoint B until it forms a straight line with the initial side, creating a straight angle ABF</note>. According to the definition of a straight angle, <note>a straight angle measures 180 degrees, i.e., angle ABF = 180 degrees</note>.


Question: As shown in the figure, AB is the diameter of ⊙O, points C and D are two points on the circle, and ∠AOC=126°, then ∠CDB=()

Answer: 27°

1. Given ∠AOC=126°, ##line segment AB is the diameter, according to the definition of a straight angle,## thus ∠AOB is a straight angle, i.e., ∠AOB=180°.

2. According to ##the definition of adjacent supplementary angles##, we get ∠BOC = ∠AOB - ∠AOC = 180° - 126° = 54°.

3. According to the inscribed angle theorem, in a circle, the measure of a central angle is twice the measure of the inscribed angle that subtends the same arc, thus ∠CDB = 1/2 ∠BOC = 1/2 × 54° = 27°.

4. Through the above reasoning, the final answer is ∠CDB=27°.

圆; 圆心角; 圆周角; 弦; 圆内接四边形

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.

In the figure of this problem, <note>circle O</note>, <note>point A and point C are two points on the circle</note>, <note>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>. Similarly, <note>the angle ∠BOC formed by the lines OB and OC is also a 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 the figure of this problem, in circle <note>O</note>, the vertex of angle <note>CDB</note>, <note>D</note>, is on the circumference, and the two sides of angle <note>CDB</note> intersect circle <note>O</note> at point <note>C</note> and point <note>B</note>, respectively. Therefore, angle <note>CDB</note> is an inscribed angle.

Definition of Linear Pair of Angles

In the figure of this problem, <note>angle AOC and angle BOC share a common side OC</note>, <note>their other sides AO and OB are extensions in opposite directions</note>, so angle AOC and angle BOC are adjacent supplementary angles.

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>AB is the diameter</note>, connecting <note>the center O</note> and <note>points A and B on the circumference</note>, with a length of <note>twice the radius</note>, i.e., <note>AO = BO</note>.

Definition of Straight Angle

The ray <note>AO</note> rotates around the endpoint <note>O</note> until it forms a straight line with the initial side, creating a <note>straight angle AOB</note>. According to the definition of a straight angle, the measure of a straight angle is 180 degrees, i.e., <note>angle AOB = 180 degrees</note>.

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>∠CDB</note> is the inscribed angle of the arc <note>CB</note>, and <note>∠BOC</note> is the central angle of the same arc <note>CB</note>. Therefore, according to the <note>Inscribed Angle Theorem</note>, <note>∠CDB = 1/2 ∠BOC</note> = 1/2 × 54° = 27°.


Question: For a pair of similar figures, use the given area to find the scale factor from the blue figure to the green figure.

Answer: \frac { 1 } { 2 }

1. ##Given that the blue figure and the green figure are similar figures, and the area of the blue figure is 8 square inches, the area of the green figure is 32 square inches.##

2. ##According to the theorem that the ratio of the areas of similar polygons is equal to the square of the similarity ratio, we have (S1/S2) = (l1/l2)^2,## where S1 and S2 are the areas of the two similar figures, and l1 and l2 are the corresponding side lengths of the two similar figures.

3. Substitute the given areas into the area ratio formula: 8/32 = (4/x)^2.

4. Calculate: 1/4 = (4/x)^2.

5. Solve the equation: (4/x)^2 = 1/4, taking the square root of both sides gives: 4/x = 1/2.

6. Solve the equation: 4/x = 1/2, multiplying both sides gives: 2*4 = x, i.e., x = 8.

7. It is deduced that x = 8, so the ##scale factor## from the blue figure to the green figure is 4/8 = 1/2.

8. Through the above reasoning, the final answer is that the ##scale factor## from the blue figure to the green figure is 1/2.

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>The blue figure and the green figure are similar figures</note>, the area of the blue figure is 8 square inches, the area of the green figure is 32 square inches, <note>the side length of the blue figure is 4 inches, corresponding to the side length of the green figure as x inches</note>, therefore we get <note>8/32 = (4/x)^2</note>.


Question: Find the scale factor from W to W'.

Answer: \frac { 1 } { 3 }

1. From the figure, it is known that the scale factor from W to W' is FW'/FW=X/12.

3. Therefore, the scale factor from W to W' is FW'/FW=4/12=1/3.

Definition of Line Segment

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

<note>Line segment FW is a part of a straight line, including endpoint F and endpoint W and all points between them</note>, with a length of 12 units. <note>Line segment FW'</note> is a part of a straight line, including endpoint F and endpoint W' and all points between them, with a length of x units. <note>Line segment W'W is a part of a straight line, including endpoint W' and endpoint W and all points between them</note>, with a length of 8 units.

Scale Factor

The scale factor refers to the ratio of the lengths of corresponding line segments of a figure after scaling to the lengths of the corresponding line segments of the original figure. If the scale factor is \( k \), then the length of each side of the scaled figure is \( k \) times the length of the corresponding side of the original figure.

<note>The triangle with vertex W'</note> is obtained by scaling <note>the triangle with vertex W by a scale factor of k</note>. According to the definition of the scale factor, <note>the length of each side of the scaled triangle is equal to k times the length of the corresponding side of the original triangle</note>. That is: <note>scaled side length = k × original side length</note>.


Question: These polygons are similar. Find x.

Answer: 6

1. Given condition: These polygons are similar. According to the definition of similar figures, the ratios of corresponding sides of similar figures are equal.

2. Therefore AG/AD = AE/AB.

3. Given DG = 7.5, AD = x, EB = 5, AB = x - 2.

4. Thus, (AD + DG)/AD = (AB + BE)/AB. Substituting the values, we get (7.5 + x)/x = (x - 2 + 5)/(x - 2). Combining the similarity ratio, we obtain the similarity ratio equation.

5. By cross-multiplying to solve the equation: (7.5 + x)*(x - 2) = x*(x + 3).

6. Expanding the equation: x^2 + 3x = 7.5x - 15 + x^2 - 2x.

7. Moving 7.5x, x^2, and -2x to the left side of the equation: 5x - 7.5x = -15.

8. Simplifying the equation: -2.5x = -15.

9. Dividing both sides by -2.5: x = -15 / -2.5.

10. Obtaining the solution for x: x = 6.

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

矩形; 位似; 平行线; 线段; 普通多边形

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>In the figure of this problem, figures EFGA and ABCD 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 AG to side AD is equal to the ratio of side AE to side AB</note>, that is, <note>AG/AD = AE/AB</note>. At the same time, their corresponding angles are also equal, that is, <note>∠AEF = ∠ABC</note>, <note>∠EFG = ∠BCD</note>, <note>∠AGF = ∠ADC</note>, <note>∠EAG = ∠BAD</note>.


Question: Find m \angle 4.

Answer: 33

1. The given condition is: an exterior angle of 123 degrees. According to the exterior angle theorem of a triangle, the exterior angle is equal to the sum of the two non-adjacent interior angles.

2. Let the triangle be BCD, the right angle of 90 degrees be ∠B, and the angle of 123° be ∠CDE.

3. According to the exterior angle theorem of a triangle, it follows that ∠CDE = ∠B + ∠4 = 123°.

####

4. Since ∠B=90°, then the equation is: 90° + ∠4 = 123°.

5. Through algebraic operations, it follows that ∠4 = 123° - 90° = 33°.

6. Through the above reasoning, the final answer is that m ∠4, i.e., the degree of ∠4, is 33°.

直角三角形; 三角形的外角; 邻补角

Definition of Exterior Angle of a Polygon

An exterior angle of a polygon is the angle formed between one side of the polygon and the extension of an adjacent side.

<note>The interior angle of polygon BCD is ∠BDC</note>, extending the adjacent sides of this interior angle <note>BD and CD forming an angle of 123° is called the exterior angle of interior angle ∠BDC</note>.

Exterior Angle Theorem of Triangle

An exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.

The <note>exterior angle of 123 degrees of the triangle</note> is equal to the sum of the two non-adjacent interior angles. <note>Suppose the right angle of 90 degrees in the triangle is ∠B, and the exterior angle of 123 degrees is ∠D, then ∠D = ∠B + ∠4, that is, 123° = 90° + ∠4</note>.


Question: As shown in the figure, AB∥CD, line EF intersects AB at point E and intersects CD at point F, EG bisects ∠BEF, intersecting CD at point G, ∠1=50°, then ∠2 equals ()

Answer: 65°

1. Given AB∥CD, according to ##Parallel Postulate 2##, the interior angles on the same side are supplementary, thus ##∠1## + ∠BEF = 180°.

2. From ∠1 = 50° and the conclusion from the previous step, we can obtain ∠BEF = 180° - 50° = 130°.

3. Since EG bisects ∠BEF, according to the ##definition of angle bisector##, we can obtain ∠BEG = ∠GEF = 1/2 * 130° = 65°.

4. Line AB is parallel to line CD, ##according to Parallel Postulate 2, alternate interior angles are equal, thus ∠2 = ∠BEG = 65°##.

5. Through the above reasoning, the final answer is 65°.

平行线; 内错角; 直线

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.

<note>The vertex of angle BEF is point E</note>, from <note>point E a line EG is drawn</note>, this line <note>divides angle BEF into two equal angles</note>, namely <note>angle BEG and angle GEF are equal</note>. Therefore, <note>line EG is the angle bisector of angle BEF</note>.

Parallel Postulate 2 of Parallel Lines

In the figure of this problem, <note>AB∥CD</note>, <note>line EF intersects these two parallel lines</note>, forming the following geometric relationship: ####Same-side interior angles:##<note>∠1 and ∠BEF</note>##are supplementary, that is##<note>∠1 + ∠BEF</note>## = 180 degrees.##Similarly, <note>AB∥CD</note>, <note>line EG intersects these two parallel lines</note>, forming the following geometric relationship: alternate interior angles: <note>∠2 and ∠BEG</note>are equal.##These relationships illustrate that when <note>two parallel lines are intersected by a third line</note>, ####alternate interior angles are equal, and same-side interior angles are supplementary.##

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.

<note>Two lines AB and CD</note> are intersected by <note>the third line EF</note>, <note>angles EFG and BEF are on the same side of the intersecting line EF and within the intersected lines AB and CD</note>, so <note>angles EFG and BEF are consecutive interior angles</note>. Consecutive interior angles EFG and BEF are supplementary, that is, <note>angle EFG + angle BEF = 180 degrees</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.

The two parallel lines <note>AB and CD</note> are intersected by a <note>line EG</note>, where <note>angle BEG and angle EGF are located between the two parallel lines and on opposite sides of the intersecting line EG</note>. Therefore, <note>angle BEG and angle EGF are alternate interior angles</note>. Alternate interior angles are equal, that is, <note>angle BEG is equal to angle EGF</note>.


Question: Find x.

Answer: 5

1. Given the condition of a right triangle, ##the hypotenuse is 10, one of the angles is 60°, according to the triangle angle sum theorem, the other angle is 30°##.

2. In a right triangle, ##according to the properties of a 30°-60°-90° triangle, the side opposite the 30° angle is half of the hypotenuse, i.e., x=10/2=5##

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

直角三角形; 正弦; 余弦; 正切

Definition of Right Triangle

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

<note>The angle is a right angle (90 degrees)</note>, so the triangle is a right triangle. <note>Sides x and y are the legs</note>, <note>side 10 is the hypotenuse</note>.

Triangle Angle Sum Theorem

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

<note>In a triangle, the three angles are the three interior angles of the triangle</note>, according to the Triangle Angle Sum Theorem, <note>the sum of the angles is 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 triangle, <note>the three angles are 30°-60°-90°</note>. <note>Side 10 is the hypotenuse</note>, <note>side x is the side opposite the 30-degree angle</note>, <note>side y is the side opposite the 60-degree angle</note>. According to the properties of a 30°-60°-90° triangle, <note>side x is half of side 10</note>, <note>side y is √3 times side x</note>. That is: <note>x = 1/2 * 10</note>, <note>y = x * √3</note>.


Question: As shown in the figure, point AC is the diameter of ⊙O. If ∠OBC=40°, then the degree of ∠AOB is ()

Answer: 80°

1. Given OC=OB, since they are two radii of the same circle, according to the properties of an isosceles triangle, we can deduce ∠OBC=∠OCB=40°.

2. In the circle, the angle subtended by arc AB at the circumference is ∠ACB, while the angle subtended at the center is ∠AOB.

3. According to the theorem of angles subtended by the same arc, the central angle is twice the angle at the circumference, so ∠AOB=2×∠OCB.

4. Since in step 1 we deduced ∠OCB=40°, therefore ∠AOB=2×40°=80°.

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

圆; 圆周角; 圆心角; 弧; 直角三角形

Definition of Diameter

<note>Point A and point C are two points on circle O</note>, <note>line segment AC passes through the center O</note>, therefore <note>AC is the diameter of circle O</note>. According to the definition of diameter, the diameter is a line segment that passes through the center and has both endpoints on the circle, it is the longest chord of the circle, with a length of 2 times the radius.

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 circle O, <note>point O is the center of the circle</note>, <note>point C and point B are any points on the circle</note>, <note>line segment OC and line segment OB are segments from the center O to any point on the circle</note>, therefore, <note>line segment OC and line segment OB are the radii of circle O</note>.

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, in triangle OBC, <note>side OC and side OB are equal</note>, they are the radii of the same circle, therefore <note>triangle OBC is an isosceles triangle</note>.

Inscribed Angle Theorem

<note>The inscribed angle subtended by arc AB is ∠ACB</note>, while <note>the central angle subtended by the same arc is ∠AOB</note>. According to the Inscribed Angle Theorem, <note>∠AOB is twice the inscribed angle ∠ACB subtended by arc AB</note>, i.e., <note>∠AOB = 2×∠ACB</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 the problem diagram, in circle O, the vertex of angle ACB (point C) is on the circumference of the circle, and the two sides of angle ACB intersect circle O at points A and B respectively. Therefore, angle ACB is an inscribed angle.

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 circle O, points A and B are two points on the circle, and the center of the circle is point O</note>. <note>The angle ∠AOB formed by the lines OA and OB is called the central angle</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 the isosceles triangle OBC, side OB and side OC are equal</note>. Therefore, according to the properties of isosceles triangles, <note>the angles opposite the equal sides are equal</note>, that is, <note>angle OBC = angle OCB</note>.


Question: Find the circumference of the figure. Round to the nearest tenth.

Answer: 19.5

1. The given condition is that the diameter of the circle is 6.2 cm.

2. According to the formula for the circumference of a circle C = π * d, where C represents the circumference and d represents the diameter, π is generally taken as 3.14.

3. Substituting the given condition, we get the circumference of the circle C = 3.14 * 6.2.

4. Calculating this, we get C = 19.468, rounding to one decimal place, we get the answer as 19.5.

5. Through the above reasoning, the final answer is 19.5 cm.

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, let the center of the circle be O</note>, <note>the line segment passing through the center intersects the circumference at points A and B, then AB is the diameter</note>, connecting the center O and points A and B on the circumference, the length is twice the radius, that is, <note>AB = 2AO</note>.

Circumference Formula of Circle

The circumference of a circle is the length around the circular boundary. It can be calculated using the formula \( C = 2\pi r \), where \( C \) represents the circumference and \( r \) represents the radius.

In the figure of this problem, <note>the diameter of the circle d is 6.2 cm</note>. According to the circumference formula of the circle <note>C = 2πr = π * d</note>, where C represents the circumference, <note>d represents the diameter</note>. Substituting the diameter into the formula, we get the circumference of the circle <note>C = 3.14 * 6.2</note>. By calculation, the specific circumference of the circle is 19.468 cm, further rounded to one decimal place, the final answer is 19.5 cm.


Question: Find J L.

Answer: 14

1. The given conditions of the problem are the three sides of triangle JKL, namely JL = x + 12, KL = 11x - 8, and JK = 7x.

2. According to the equality of sides, we have: JK = JL = KL, i.e., x + 12 = 11x - 8 = 7x.

3. Solving the equation, we get x = 2.

4. Therefore, JL = x + 12 = 14.

5. The final answer is that the length of side JL is 14.

等腰三角形; 线段; 三角形的外角

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>Triangle JKL</note> is a geometric figure composed of <note>three non-collinear points J, K, L</note> and their <note>connecting line segments JK, KL, JL</note>. <note>Points J, K, L</note> are the <note>three vertices</note> of the triangle, and <note>line segments JK, KL, JL</note> are the triangle's <note>three sides</note>.

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 JKL is an equilateral triangle. <note>Side JK</note>, <note>side KL</note>, and <note>side JL are equal in length, all being 14</note>, and <note>angle JKL</note>, <note>angle KLJ</note>, and <note>angle LJK are equal in degree, all being 60°</note>.


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

Answer: 8.5

1. Given that the segment VU is a tangent to the circle, the center of the circle is point T, and the point of tangency is U, TV = 11. According to the property of the tangent to a circle, the radius UT is perpendicular to VU.

2. According to the definition of a right triangle, triangle UVT is a right triangle. Using the Pythagorean theorem (the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides), which applies to the right triangle UVT, therefore VU^2 = VT^2 - UT^2.

3. Given UV = 7, VT = 11, let UT = x. Substitute into the Pythagorean theorem formula: 7^2 = 11^2 - x^2, which can be transformed into the equation: 49 = 121 - x^2.

4. Simplify the equation and solve for the square root: 49 = 121 - x^2 => x^2 = 121 - 49 => x^2 = 72 => x = √72 ≈ 8.485.

5. Through the above reasoning, the answer rounded to one decimal place is 8.5.

切线; 直角三角形; 圆

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.

The original text: <note>The circle and ##line segment VU## have only one ##common point U##</note>, this common point is called the tangent point. Therefore, <note>##line segment VU## is the tangent to the circle</note>.

Definition of Right Triangle

<note>In the figure of this problem, in triangle VUT, angle ∠VUT is a right angle (90 degrees), so triangle VUT is a right triangle.</note><note>Side VU</note> and <note>side UT are the legs</note>, <note>side VT is the hypotenuse</note>.

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.

<note>Point U</note> is the point of tangency between <note>line VU</note> and the circle, and <note>segment TU</note> is the radius of the circle. According to the property of the tangent line to a circle, <note>tangent line VU</note> is perpendicular to the <note>radius TU</note> at the <note>point of tangency U</note>, i.e., <note>∠VUT</note> = 90 degrees.

Pythagorean Theorem

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

<note>Triangle VUT is a right triangle</note>, <note>∠VUT is a right angle (90 degrees)</note>, side UT and <note>side VU are the legs</note>, <note>side VT is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>VT^2 = UT^2 + VU^2</note>.


Question: Find x.

Answer: 12.5

1. Given conditions: In triangle WXY, ZT is parallel to XY, where XY = 16, ZT = 10, and WY = 20.

2. According to the parallel axiom 2 of parallel lines, ∠WZT = ∠X, and since ∠W is a common angle, according to the AA criterion of similar triangles, triangle TZW is similar to triangle WXY.

3. According to the properties of similar triangles:

4. Based on the proportional relationship of similar triangles and the following calculation:

5. Through the above reasoning, the final answer is 12.5.

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 WZT and triangle WXY are similar triangles</note>. According to the definition of similar triangles: <note>∠W = ∠W</note>, <note>∠WZT = ∠WXY</note>, <note>∠WTZ = ∠WYX</note>; <note>ZT/XY = WT/WY</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.

<note>In the figure of this problem, two parallel lines ZT and XY</note> are <note>intersected by a third line WZ</note>, forming the following geometric relationship: 1. <note>Corresponding angles: angle WZT and angle X</note> are equal.

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.

In the figure of this problem, <note>two parallel lines ZT and XY</note> are intersected by <note>a transversal line WX</note>, where <note>angle WZT and angle X are on the same side of the transversal WX</note>, <note>on the same side of the two intersected lines ZT and XY</note>. Therefore, <note>angle WZT and angle X are corresponding angles</note>. <note>Corresponding angles are equal</note>, that is, <note>angle WZT is equal to angle X</note>.

Similarity Theorem for Triangles (AA)

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

<note>Angle W is equal to angle W</note>, and <note>angle WZT is equal to angle X</note>, so <note>triangle WZT is similar to triangle WXY</note>.


Question: Solve for x.

Answer: 9.6

1. Given that there is a large triangle and a small triangle in the figure, ##let the vertices of the large triangle be A, B, C, and let the vertices of the small triangle be A, D, E##.

2. ##From the figure, we can see that angle A is the common angle of the large triangle ABC and the small triangle ADE, i.e., ∠DAE=∠BAC. DE is parallel to BC. According to the parallel postulate 2 of parallel lines, corresponding angles are equal, so ∠ADE=∠ABC##.

3. ##From the previous step, we get ∠DAE=∠BAC and ∠ADE=∠ABC. According to the AA similarity theorem, the large triangle ABC is similar to the small triangle ADE##.

4. ##According to the definition of similar triangles, corresponding sides are proportional, i.e., AD/AB=AE/AC##.

5. ##From the figure, we know that AD=x, AB=4+x, AE=12, AC=12+5. Substituting into the proportion equation, we get: x/(4+x)=12/(12+5)##.

6. ##Cross-multiplying, we get (4+x) * 12 = 17 * x. Simplifying, we get 48+12x=17x. Rearranging, we get 5x=48, so x=9.6##.

7. Through the above reasoning, the final answer is 9.6.

普通三角形; 等腰三角形

Definition of Similar Triangles

<note>The large triangle and the small triangle are similar</note>. According to the definition of similar triangles, there is: <note>The large triangle and the small triangle have a common angle</note>, and <note>Two angles of the large triangle are respectively equal to two angles of the small triangle</note>. According to the properties of similar triangles, <note>Corresponding sides are proportional, that is, AD/AB=AE/AC</note>.

Similarity Theorem for Triangles (AA)

<note>Angle DAE is equal to angle BAC</note>, and <note>angle ADE is equal to angle ABC</note>, so <note>triangle ABC is similar to triangle ADE</note>.

Parallel Postulate 2 of Parallel Lines

<note>In the diagram of this problem, two parallel lines DE and BC are intersected by a third line AB</note>, forming the following geometric relationship: <note>corresponding angles: angle ADE and angle ABC are equal</note>.

Definition of Corresponding Angles

In the figure of this problem, <note>two parallel lines DE and BC</note> are intersected by <note>a line AB</note>, where <note>angle ADE and angle ABC are on the same side of the intersecting line AB</note>, <note>on the same side of the intersected lines DE and BC</note>, therefore <note>angle ADE and angle ABC are corresponding angles</note>. Corresponding angles are equal, that is, <note>angle ADE is equal to angle ABC</note>.


Question: As shown in the figure, line AB ∥ CD, AE bisects ∠CAB, ∠ACD = 40°, then the degree of ∠AEC is ()

Answer: 70°

1. Given that line AB is parallel to line CD, according to the ##Parallel Postulate 2 and the definition of consecutive interior angles##, we get ∠CAB + ∠ACD = 180°.

2. From the given condition ∠ACD = 40°, and the conclusion from the previous step, we get ∠CAB = 180° - 40° = 140°.

3. According to the problem statement, AE bisects ∠CAB, according to the ##definition of angle bisector, we get ∠CAE = (1/2)∠CAB = (1/2) × 140° = 70°##.

4. Therefore, according to the triangle angle sum theorem, in triangle AEC, ##∠AEC = 180°-∠CAE-∠ACE = 180°-70°-40°=70°##.

5. Through the above reasoning, the final answer is 70°.

平行线; 内错角; 同旁内角; 线段; 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 CAB is point A</note>, a line AE is drawn from <note>point A</note>, which divides angle CAB into two equal angles, that is, <note>angle BAE and angle EAC are equal</note>. Therefore, <note>line AE is the angle bisector of angle CAB</note>.

Parallel Postulate 2 of Parallel Lines

In the figure of this problem, <note>line AB is parallel to line CD</note>, <note>line AC intersects these two parallel lines</note>, forming the following geometric relationships: ##1. Consecutive interior angles: <note>∠CAB and ∠ACD are supplementary</note>, that is, <note>∠CAB + ∠ACD = 180 degrees</note>##. These relationships illustrate that when <note>two parallel lines are intersected by a third line</note>, corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary.##

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.

In the figure of this problem, <note>two lines AB and CD</note> are intersected by <note>a third line AC</note>, <note>forming angles BAC and ACD on the same side of the transversal AC</note>, and within the intersected lines <note>AB and CD</note>. Therefore, <note>angles BAC and ACD are consecutive interior angles</note>. <note>Consecutive interior angles BAC and ACD are supplementary</note>, that is, <note>angle BAC + angle ACD = 180 degrees</note>.

Triangle Angle Sum Theorem

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

<note>Angles ∠CAE, ∠AEC, and ∠ACE are the three interior angles of triangle AEC</note>, according to the Triangle Angle Sum Theorem, <note>∠CAE + ∠AEC + ∠ACE = 180°</note>.


Question: Find the degree measure of \angle 1.

Answer: 65

1. By drawing a right angle mark at the intersection of lines EG and DG in the figure, it indicates that these two lines are perpendicular to each other, so ∠EGD is a right angle, i.e., ∠EGD=90°. In triangle DEG, it is known that ∠EGD=90° and ∠E=25°.

2. According to the triangle angle sum theorem, the sum of the three interior angles of any triangle is 180°, so ∠E+∠1+∠EGD=180°. Given that ∠EGD=90° and ∠E=25°, therefore ∠1=180°-∠E-∠EGD=180°-90°-25°=65°.

3. Through the above reasoning, the final answer is ∠1=65°.

直角三角形; 三角形的外角

Definition of Triangle

<note>Triangle EGD</note> is a geometric figure composed of <note>three non-collinear points E, G, D</note> and their connecting <note>line segments EG, ED, DG</note>. Points <note>E, G, D are the three vertices of the triangle</note>, and <note>line segments EG, ED, DG are the three sides of the triangle</note>.

Triangle Angle Sum Theorem

<note>In triangle EDG, angle GED, angle EGD, and angle EDG are the three interior angles of triangle EFG</note>, according to the Triangle Angle Sum Theorem, <note>angle GED + angle EGD + angle EDG = 180°</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>The angle ∠DGE formed by the intersection of line DG and line EF is 90 degrees</note>, therefore according to the <note>Definition of Perpendicular Lines</note>, <note>line DG and line EF are perpendicular to each other</note>.


Question: Find the area of the shaded region. Round to the nearest tenth.

Answer: 157.1

1. Observe the figure and notice that it is a large circle with a diameter of 20, containing two smaller circles of the same radius, and the smaller circles are tangent to the horizontal diameter at the same point.

2. According to the given conditions, the diameter of the large circle is 20, so the radius of the large circle is 20/2 = 10.

3. Divide the diameter of 20 into two parts, each part being the diameter of the smaller circle. From the information in the figure, it is inferred that each smaller circle has a diameter of 10. Therefore, the radius of the smaller circle is 10/2 = 5.

4. The area formula for the large circle is A = \\pi \\times r^2, so the area of the large circle is A_large = \\pi \\times 10^2 = 100\\pi.

5. Similarly, the area formula for each smaller circle is A_small = \\pi \\times 5^2 = 25\\pi. Since there are two smaller circles in the figure, the total area of the two smaller circles is 50\\pi.

6. The area of the shaded part (annular region) is the area of the large circle minus the area of the two smaller circles, so the area of the annular region is 100\\pi - 50\\pi = 50\\pi.

7. Using \\pi \\approx 3.14159, the area of the annular region is 50 \\times 3.14159 = 157.0795. According to the problem requirements, the final result should be rounded to one decimal place.

8. After calculation and rounding, the area of the annular region is 157.1 (rounded to one decimal place).

Definition of Radius

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

<note>In the diagram of this problem, in the large circle, the center is the center point of the large circle, any point on the circle is any point on the circumference of the large circle, and the distance from the center to any point on the circumference is the radius of the large circle</note>, therefore the radius of the large circle is 10 (derived from the diameter of the large circle being 20, 20/2=10). <note>In each small circle, the center is the center point of the small circle, any point on the circle is any point on the circumference of the small circle, and the distance from the center to any point on the circumference is the radius of the small circle</note>, therefore the radius of each small circle is 5 (derived from the diameter of the small circle being 10, 10/2=5).

Definition of Diameter

<note>The diameter of the large circle</note> is 20, connecting the center of the circle and two points on the circumference, the length is 2 times the radius, that is, 20 = 2 * 10. <note>The diameter of each small circle</note> is 10, connecting the center of the small circle and two points on the circumference, the length is 2 times the radius of the small circle, that is, 10 = 2 * 5.

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, the radius of the large circle is <note>10</note>, and the radius of the small circle is <note>5</note>. According to the area formula of a circle, the area of a circle A equals the circumference π multiplied by the square of the radius. The area of the large circle is <note>A_large = π × 10^2 = 100π</note>, and the area of the small circle is <note>A_small = π × 5^2 = 25π</note>.

Definition of Tangent Circles

Two circles are tangent if they intersect at exactly one point, and their tangents at the point of contact coincide.

The original text: <note>The two small circles above and below are tangent to the large circle at a common point</note>, which means that the circumference of each small circle is tangent to the circumference of the large circle at <note>this one common point</note>.


Question: Find x. Round to the nearest tenth if necessary. Assume that the segments that appear tangent are tangent.

Answer: 2

1. Let two intersecting chords within a circle be AB and CD, and the intersection point of AB and CD be point E. Given that chords AB and CD intersect at point E within the circle, according to the intersecting chords theorem, we can obtain AE * EB = CE * ED.

2. Given AE=5, EB=4, CE=x, ED=x + 8, substituting into AE * EB = CE * ED, we get 5 * 4 = x * (x + 8). Expanding the formula, we obtain 20 = x^2 + 8x.

3. Transform the above equation into x^2 + 8x - 20 = 0 for factorization, resulting in (x + 10)(x - 2) = 0.

4. Therefore, we can obtain two roots, x = -10 or x = 2.

5. Since lengths in geometric figures cannot be negative, we discard x = -10.

6. Through the above reasoning, we finally obtain the answer x = 2.

Definition of Chord

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

<note>Point A and point B are any two points on the circle</note>, <note>line segment AB connects these two points</note>, so line segment AB is a chord of the circle; <note>point C and point D are any two points on the circle</note>, <note>line segment CD connects these two points</note>, so line segment CD is a chord of the circle.

Intersecting Chords Theorem

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

<note>Two chords AB and CD intersect at point E inside a circle</note>, <note>chord AB is divided into two segments AE and EB by point E</note>, <note>chord CD is divided into two segments CE and ED by point E</note>, according to the Intersecting Chords Theorem, we can conclude that <note>the length of segment AE multiplied by the length of segment EB is equal to the length of segment CE multiplied by the length of segment ED</note>.


Question: ΔLMN ≅ ΔQRS. Find x.

Answer: 5

1. Given the condition △LMN ≅ △QRS, according to the definition of congruent triangles, corresponding angles are equal and corresponding sides are equal, we can obtain LN = SQ, therefore 3x + 5 = 2x + 10.

2. Solve the equation 3x + 5 = 2x + 10, we get: x = 5.

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

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>△LMN ≌ △QRS</note>, according to the properties of congruent triangles, it is known that: <note>∠LNM = ∠QSR</note>, <note>∠LMN = ∠QRS</note>, <note>∠MLN = ∠RQS</note>, and the corresponding sides are equal, namely: <note>side LN = side QS</note>, <note>side LM = side QR</note>, <note>side MN = side RS</note>.


Question: m \angle B A C = 38, B C = 5, 和 D C = 5, 求 m \angle D A C.

Answer: 38

1. Given the condition ∠BAC=38°, according to the figure and definition, ∠ABC and ∠ADC are both right angles, i.e., ∠ABC=90°, ∠ADC=90°.

2. In △ABC, given side BC=5 and angle ∠ABC=90°, according to the definition of a right triangle, △ABC is a right triangle. BC and AB are the legs, and AC is the hypotenuse.

3. Similarly, in △ADC, given side DC=5, angle ∠ADC=90°, according to the definition of a right triangle, △ADC is a right triangle. DC and AD are the legs, and AC is the hypotenuse.

4. According to the right triangle congruence criterion (hypotenuse, leg), since BC=DC=5 and AC=AC, right triangle ABC is congruent to right triangle ADC.

5. According to the definition of congruent triangles, ∠DAC=∠BAC. Therefore, ∠DAC=38°.

6. Through the above reasoning steps, the final ∠DAC equals 38°.

普通三角形; 等腰三角形; 垂线

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.

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

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 ABC, <note>angle ∠ABC is a right angle (90 degrees)</note>, so triangle ABC is a right triangle. Side AB and side BC are the legs, and side AC is the hypotenuse. Similarly, in triangle ADC, <note>angle ∠ADC is a right angle (90 degrees)</note>, so triangle ADC is a right triangle. Side AD and side DC are the legs, and side AC is the hypotenuse.

Right Triangle Congruence (Hypotenuse, One Leg)

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

In the figure of this problem, <note>triangle ABC and triangle ADC are congruent triangles</note>, <note>the corresponding sides and angles of triangle ABC are equal to those of triangle ADC</note>, namely:
<note>side AB = side AD</note>,
<note>side BC = side CD</note>,
<note>side AC = side AC</note>,
and the corresponding angles are also equal:
<note>angle ABC = angle ADC</note>,
<note>angle ACB = angle ACD</note>,
<note>angle DAC = angle BAC</note>.


Question: Find x.

Answer: 8

1. Let the angle between the hypotenuses of the left triangle be A, the angle marked with a single arc be B, the angle marked with double arcs be C, and the midpoint of the base BC be D. Let the angle between the hypotenuses of the right triangle be X, the angle marked with a single arc be Y, the angle marked with double arcs be Z, and the midpoint of the base YZ be W.

2. From the figure, we can obtain BC=12, AD=x, YZ=15, XW=10, angle ABC=angle XYZ, angle ACB=angle XZY, and BD=CD=1/2 BC, YW=ZW=1/2 YZ. Therefore, according to the similarity theorem (AA), triangle ABC~triangle XYZ.

3. According to the definition of similar triangles: BC/YZ = AD/XW.

4. Substitute the corresponding values into the formula for similar triangles, 12/15 = x/10.

5. Solve the proportion to find x: 12/15 = x/10.

6. By cross-multiplying, we get: 12 * 10 = 15 * x.

8. Divide both sides by 15 to find x: x = 120/15.

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

Definition of Similar Triangles

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

<note>In the diagram of this problem, triangle ABC and triangle XYZ are similar triangles. According to the definition of similar triangles: angle ABC = angle XYZ, angle ACB = angle XZY, angle BAC = angle YXZ; AB/XY = BC/YZ = AD/XW.</note>

Similarity Theorem for Triangles (AA)

<note>Angle ABC is equal to angle XYZ</note>, and <note>angle ACB is equal to angle XZY</note>, so triangle ABC is similar to triangle XYZ.


Question: As shown in the figure, points A, B, C are three points on ⊙O. If ∠BOC=80°, then the degree of ∠A is ()

Answer: 40°

1. Given that points A, B, and C are three points on ⊙O, ∠BOC = 80°.

2. According to the inscribed angle theorem, the inscribed angle is half of the central angle it subtends, i.e., the inscribed angle BAC corresponding to the central angle BOC is half of it.

3. By the inscribed angle theorem: ∠BAC = 1/2 * ∠BOC.

5. Through the above reasoning, the final answer is 40°.

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 the figure of this problem, in circle O, <note>the vertex A of angle BAC is on the circumference</note>, <note>the two sides of angle BAC intersect circle O at points B and C respectively</note>. Therefore, <note>angle BAC is an inscribed angle</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 diagram of this problem, in circle O, points B and C are two points on the circle, and the center of the circle is point O</note>. <note>The angle ∠BOC formed by the lines OB and OC is called the central angle</note>.

Inscribed Angle Theorem

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

In <note>circle O</note>, <note>points A, B, and C are on the circle</note>, <note>the central angle corresponding to arc BC is ∠BOC, and the inscribed angle is ∠BAC</note>. According to the Inscribed Angle Theorem, <note>∠BAC is equal to half of the central angle ∠BOC corresponding to arc BC</note>, that is, <note>∠BAC = 1/2 ∠BOC</note>.


Question: As shown in the figure, points A, B, and C are on ⊙O and AB=AC. Connect BO and CO. If ∠ABC=65°, then the degree of ∠BOC is ()

Answer: 100°

1. Given AB=AC, points A, B, and C are on ⊙O, ##according to the definition of an isosceles triangle, we know that triangle ABC is an isosceles triangle##.

2. ##According to the properties of an isosceles triangle, we know ∠ABC=∠ACB. Also, according to the triangle angle sum theorem, we have## ∠ABC + ∠ACB + ∠BAC = 180°.

3. Given ∠ABC = 65°, ∠ACB = 65°, according to the above step 2, we can obtain ∠BAC = 180° - 65° - 65° = 50°.

4. ∠BOC is the central angle corresponding to AB and AC in ⊙O, according to the ##inscribed angle## theorem: the inscribed angle is half of the corresponding central angle. Therefore, ∠BOC = 2 * ∠BAC.

5. From step 3, we get ∠BAC=50°, thus ∠BOC=2 * 50° = 100°.

6. Through the above reasoning, the final answer is that the measure of ∠BOC is 100°.

等腰三角形; 圆; 圆周角; 圆心角

Definition of Isosceles Triangle

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

In the diagram of this problem, in triangle ABC, <note>side AB is equal to side AC</note>, therefore <note>triangle ABC is an isosceles triangle</note>.

Definition of Inscribed Angle

In the figure of this problem, in circle O, <note>the vertex B of angle ABC is on the circumference</note>, <note>the two sides of angle ABC intersect circle O at points A and C respectively</note>. Therefore, angle ABC is an inscribed angle.

Definition of Central Angle

<note>In circle O, points B and C are two points on the circle, and the center of the circle is point O</note>. The angle formed by the lines <note>OB and OC</note>, <note>∠BOC</note>, is called the central angle.

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 the figure of this problem, in the isosceles triangle ABC, side AB and side AC are equal.</note> Therefore, according to the properties of isosceles triangles, the angles opposite the equal sides are equal, i.e., <note>angle ABC = angle ACB</note>.

Triangle Angle Sum Theorem

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

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

Inscribed Angle Theorem

<note>∠BOC is the central angle corresponding to arc BC</note>, <note>∠BAC is the inscribed angle corresponding to arc BC</note>. According to the Inscribed Angle Theorem, the inscribed angle is equal to half of the central angle corresponding to the arc it intercepts, that is, <note>∠BAC = 1/2 ∠BOC</note>. Therefore, <note>∠BOC = 2 * ∠BAC = 2 * 50° = 100°</note>.


Question: Find x such that a \parallel b.

Answer: 14

1. According to the parallel axiom 2 of parallel lines and the definition of same-side interior angles, when angle (4x + 11)° and angle (8x + 1)° are supplementary, a∥b.

2. According to the definition of same-side interior angles, derive the equation: (4x + 11)° + (8x + 1)° = 180°.

3. Solve the equation: (4x + 11)° + (8x + 1)° = 180° simplified to 12x° + 12° = 180°.

4. Subtract 12° from both sides to get 12x° = 168°.

5. Divide both sides by 12° to get x° = 14°.

6. Through the above reasoning, when x = 14, a is parallel to b, and the final answer is obtained.

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.

<note>Angle (4x + 11)°</note> and <note>Angle (8x + 1)°</note> are consecutive interior angles of line <note>a</note> and line <note>b</note>. Line <note>a</note> and line <note>b</note> are intersected by a transversal, and both angles are located on the interior side of these two lines. Therefore, <note>Angle (4x + 11)°</note> and <note>Angle (8x + 1)°</note> are consecutive interior angles. Consecutive interior angles are supplementary, that is, <note>Angle (4x + 11)° + Angle (8x + 1)° = 180°</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.

<note>Angles (4x + 11)° and (8x + 1)° are the same-side interior angles formed by two lines a and b intersected by a transversal</note>. These two angles are supplementary, that is, <note>(4x + 11)° + (8x + 1)° = 180°</note>. According to the same-side interior angle theorem, it can be concluded that <note>a and b are parallel</note>.


Question: Find the area of the parallelogram.

Answer: 552

1. The figure provided in the problem is a parallelogram, with one side length of 23 feet, ##and the height on this side is 24 feet##.

2. ##According to the area formula of a parallelogram,## the formula for calculating the area of a parallelogram is: Area = base * height.

3. Therefore, according to the ##area formula of a parallelogram##, the area of the parallelogram can be calculated as 23 feet * 24 feet = 552 square feet.

4. Thus, it can be inferred that the area of the parallelogram is 552 square feet.

平行四边形; 平行线; 垂线

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 ABCD is a parallelogram</note>, <note>side AB is parallel and equal to side CD</note>, <note>side AD is parallel and equal to side BC</note>.

Area Formula of a Parallelogram

The area of a parallelogram is calculated by multiplying the length of its base by the corresponding height, which can be expressed as \( A = \text{base length} \times \text{height} \).

The original text: <note>The length of the base is 23 feet</note>, <note>The corresponding height is the vertical distance from the base to the opposite side, which is 24 feet</note>. Therefore, according to the area formula of a parallelogram, the area of the parallelogram is equal to the base length multiplied by the corresponding height, that is, <note>A = 23 feet × 24 feet = 552 square feet</note>.

Height of a Parallelogram

The height (or altitude) of a parallelogram is the perpendicular distance from a vertex on one side to the line containing the opposite side.

<note>In parallelogram ABCD, sides AB and CD are parallel</note>. <note>The perpendicular distance from vertex A to the opposite side CD is the height of the parallelogram</note>. Specifically, <note>draw a line segment from vertex A perpendicular to side CD</note>, the length of this line segment is the height of the parallelogram.


Question: Find T Q.

Answer: 12

1. Given RS = 7, ST = 7, therefore RS = ST, S is the midpoint of RT. Also, QS ⊥ QT, the intersection point is the midpoint S of RT. So according to the definition of the perpendicular bisector, QS is the perpendicular bisector of QT.

2. Since point Q is on QS, according to the property of the perpendicular bisector, QR = QT.

3. Therefore, we have x + 3 = 2x - 6, after rearranging, we get x equals 9.

4. Substituting x = 9 into 2x - 6, we get QT = 12.

等腰三角形; 垂线; 直角三角形

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 RT is point S</note>. According to the definition of the midpoint of a line segment, <note>point S divides line segment RT into two equal parts</note>, that is, the lengths of line segments RS and ST are equal. That is, <note>RS = ST</note>.

Definition of Perpendicular Bisector

A line is a perpendicular bisector if and only if it passes through the midpoint of a line segment and is perpendicular to that line segment.

The line QS passes through the midpoint S of the line segment RT, and the line QS is perpendicular to the line segment RT. Therefore, the line QS is the perpendicular bisector of the line segment RT.

Properties of the Perpendicular Bisector

Any point on the perpendicular bisector of a line segment is equidistant from the endpoints of the segment.

In the figure of this problem, <note>the perpendicular bisector of segment RT is line QS</note>, <note>point Q is on line QS</note>. According to the properties of the perpendicular bisector, <note>the distances from point Q to the endpoints R and T of segment RT are equal, that is, QR = QT</note>.


Question: If c = 5, find b.

Answer: 2.5 \sqrt { 3 }

1. Given conditions: In the right triangle ABC, ∠C = 90°, ∠B = 60°, ∠A = 30°, hypotenuse c = 5.

2. According to the properties of a 30°-60°-90° triangle, the side opposite the 30° angle is half of the hypotenuse, and the side opposite the 60° angle is √3 times the shorter side.

3. Given that the hypotenuse AB(c) = 5, then BC(a) = 5/2 = 2.5.

4. Since AC(b) is the side opposite the 60° angle, AC(b) = 2.5 * √3.

5. Therefore, b = 2.5 * √3.

6. Through the above reasoning, the final answer is 2.5 * √3.

直角三角形; 正弦; 余弦

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 ABC, <note>angle ACB is a right angle (90 degrees)</note>, therefore <note>triangle ABC is a right triangle</note>. <note>Side AC and side BC are the legs, and side AB is the hypotenuse</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.

<note>30°-60°-90° Triangle ABC</note> has <note>angle BAC as 30 degrees</note>, <note>angle CBA as 60 degrees</note>, and <note>angle BCA as 90 degrees</note>. <note>Side AB is the hypotenuse</note>, <note>side BC is opposite the 30-degree angle</note>, and <note>side AC is opposite the 60-degree angle</note>. According to the <note>properties of 30°-60°-90° Triangle</note>, <note>side BC is half of side AB</note>, and <note>side CA is √3 times side BC</note>. That is: <note>BC = 1/2*AB</note>, <note>CA = BC *√3</note>.


Question: Find m \angle 1.

Answer: 108

1. In △RQP, using the triangle angle sum theorem, we get ∠RQP + ∠QPR + ∠PRQ = 180°.

2. According to the given conditions in the problem, ∠PQR = 90°, and it is also known that ∠QRP = 33°.

3. Substitute the above known conditions into the angle sum formula: 90° + ∠QPR + 33° = 180°.

4. Simplifying the equation, we get ∠QPR = 180° - 90° - 33° = 57°.

5. In △QPS, ∠PQS = 51°, ∠QPS = 57°, using the triangle angle sum theorem, we get ∠QSP = 180° - ∠SQP + ∠QPS = 180° - 57° - 51° = 72°.

6. According to the definition of supplementary angles, ∠1 = ∠QSR, we get ∠1 as the supplementary angle of ∠QSP, ∠1 = 180° - ∠QSP = 180° - 72° = 108°.

7. Based on the information obtained from the above steps, we finally get ∠1 = 108°.

直角三角形; 三角形的外角; 内错角

Triangle Angle Sum Theorem

<note>In triangle RQP, angle RQP, angle QPR, and angle PRQ are the three interior angles of triangle RQP</note>, according to the Triangle Angle Sum Theorem, <note>angle RQP + angle QPR + angle PRQ = 180°</note>. In triangle PQS, <note>angle PQS, angle QPS, and angle PSQ are the three interior angles of triangle PQS</note>, according to the Triangle Angle Sum Theorem, <note>angle PQS + angle QPS + angle PSQ = 180°</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.

<note>Angle QSP and angle QSR share a common side QS</note>, and their other sides <note>PS and RS are extensions in opposite directions</note>, so <note>angle QSP and angle QSR are adjacent supplementary angles</note>.


Question: As shown in the figure, Xiaoying designed a method to measure the height of an ancient city wall using a flashlight. A horizontal plane mirror is placed at point P, and the light ray from point A is reflected by the plane mirror and just reaches the top of the ancient city wall at point C. It is known that AB⊥BD, CD⊥BD, and the measurements are AB=1.4 meters, BP=2.1 meters, PD=12 meters. Then the height of the ancient city wall CD is ()

Answer: 8米

1. Given that the light ray starts from point A and is reflected by the plane mirror P, just reaching the top of the ancient city wall CD at point C, according to the ##law of reflection, the angle of incidence equals the angle of reflection##, we get ∠APB=∠CPD.

2. Given AB⊥BD and CD⊥BD, i.e., ∠ABD=∠CDB=90°, we deduce ∠ABP=∠CDP.

3. By the ##similar triangles criterion (AA)##, from ∠APB=∠CPD and ∠ABP=∠CDP, we get ##triangle ABP ∽ triangle CDP##.

4. According to the ##definition of similar triangles##, the corresponding sides of similar triangles are proportional, we get ##AB/CD=BP/PD##.

5. Given the conditions AB=1.4 meters, BP=2.1 meters, PD=12 meters, substitute into the proportional relationship ##1.4/CD=2.1/12##.

6. Solving the proportion equation, we get CD=8 meters.

7. Through the above reasoning, the final answer is 8 meters.

反射; 直角三角形; 垂线; 平行线

Law of Reflection

When a ray of light reflects, the angle of incidence is equal to the angle of reflection, and the reflected ray lies on the same plane as the incident ray and the normal.

<note>The light ray shoots from point A to the reflective surface P</note>, and <note>reflection occurs at point P</note>. <note>The incident angle ∠APB is the angle between the light ray AP and the normal</note>, and <note>the reflection angle ∠CPD is the angle between the reflected light ray PC and the normal</note>. According to the law of reflection, <note>the incident angle ∠APB is equal to the reflection angle ∠CPD</note>, and <note>the reflected light ray PC is on the same side of the plane as the normal and the incident light ray AP</note>.

Similarity Theorem for Triangles (AA)

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

In the original text: <note>Triangle ABP and triangle CDP</note>, if <note>angle ABP is equal to angle CDP</note>, and <note>angle APB is equal to angle CPD</note>, then <note>triangle ABP is similar to triangle CDP</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, triangles <note>ABP</note> and <note>CDP</note> are similar triangles. According to the definition of similar triangles, we have: <note>∠APB = ∠CPD</note>, <note>∠ABP = ∠CDP</note>, <note>∠PAB = ∠PCD</note>; <note>AB/CD=BP/PD</note>.


Question: As shown in the figure, in Rt△ABC, ∠C=90°, AC=4, AB=5, then the value of sinB is ()

Answer: \frac{4}{5}

1. Given that in right triangle ABC, ∠ACB=90°, AC=4, AB=5.

2. According to the definition of the sine function, sin∠ABC = opposite side AC/hypotenuse AB.

3. Substituting the side lengths, sin∠ABC = 4/5.

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

直角三角形; 正弦; 线段

Definition of Right Triangle

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

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.

<note>In right triangle ABC</note>, <note>angle B is an acute angle</note>, <note>side AC is the opposite side of angle B</note>, <note>side AB is the hypotenuse</note>. According to the definition of the sine function, <note>the sine of angle B is equal to the ratio of the opposite side AC to the hypotenuse AB</note>, that is, <note>sin∠ABC = AC / AB</note>.


Question: Express the value of \cos A as a decimal, accurate to two decimal places.

Answer: 0.6

1. Given △ABC is a right triangle, ∠ABC is a right angle, AB=15, BC=20, and AC=25.

2. In a right triangle, the hypotenuse is the longest side. Here, AC is the hypotenuse.

####

##3. According to the cosine function##, in a right triangle, the cosine ratio of an acute angle is the ratio of the length of the adjacent side to the length of the hypotenuse.

##4##. Since we need to find cos∠CAB, we should compare sides AB and AC. cos A is the cosine ratio of ∠CAB.

##5. That is## cos∠CAB = adjacent side AB/hypotenuse AC = 15/25.

##6##. Simplifying the ratio 15/25, we get cos∠CAB = 3/5.

##7##. Converting the simplified ratio 3/5 to a decimal, the result is 0.6.

##8##. Through the above reasoning, the final answer is 0.60.

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

Definition of Right Triangle

<note>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>.

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 figure of this problem, <note>right triangle ABC</note>, <note>side AB is the adjacent side of angle ∠CAB</note>, <note>side AC is the hypotenuse</note>. Therefore, according to the definition of the cosine function, <note>the cosine value of angle ∠CAB is equal to the ratio of the adjacent side AB to the hypotenuse AC, that is, cos(∠CAB) = AB / AC</note>.


Question: Find x such that the quadrilateral is a parallelogram.

Answer: 5

1. Given a quadrilateral is a parallelogram ####.

2. Let the four vertices of the parallelogram be ABCD, and the diagonals intersect at point O.

3. According to the properties of the parallelogram, we have AO = CO, BO = DO.

4. From the figure, we know: AO = x + 4, CO = 3x - 6.

####.

5. Since AO = CO, we have the following equation:

   x + 4 = 3x - 6

6. Solving the equation x + 4 = 3x - 6, rearranging terms gives 4 + 6 = 3x - x, resulting in 2x = 10, solving for x gives x = 5.

7. Therefore, the value of x is 5.

平行四边形; 内错角; 对顶角; 同旁内角; 邻补角

Properties of Parallelogram Theorem

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

In the <note>parallelogram ABCD</note>, the opposite angles <note>\angle ABC and \angle CDA are equal</note>, and the opposite angles <note>\angle BCD and \angle DAB are equal</note>; the sides <note>AB and CD are equal</note>, and the sides <note>AD and BC are equal</note>; the diagonals AC and BD <note>bisect each other</note>, that is, the <note>intersection point O</note> divides the diagonal AC into two equal segments <note>AO and CO</note>, and divides the diagonal BD into two equal segments <note>BO and DO</note>.


Question: Find the area of the trapezoid.

Answer: 678.5

1. Given that the top side of the trapezoid is 22 ft, the bottom side is 37 ft, and the height is 23 ft. According to the formula for the area of a trapezoid, the area is equal to 1/2 multiplied by the sum of the top and bottom sides, then multiplied by the height.

2. Substitute the values: Area = 1/2 * (22 ft + 37 ft) * 23 ft.

3. Calculate the sum inside the parentheses: 22 ft + 37 ft = 59 ft.

4. Substitute the result into the formula: Area = 1/2 * 59 ft * 23 ft.

5. Calculate 59 multiplied by 23, the result is 1357.

6. Then divide by 2 to get the area of the trapezoid: 1357/2 = 678.5.

7. Therefore, the area of the trapezoid is 678.5 square feet.

Definition of Trapezoid

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

<note>The upper base of the trapezoid is 22 feet, representing the parallel upper side</note>, <note>the lower base is 37 feet, representing the parallel lower side</note>. According to the definition of a trapezoid, a trapezoid is a quadrilateral with exactly one pair of parallel sides.

Definition of Altitude

An altitude is a perpendicular line segment extending from a vertex of a triangle to the opposite side (or to the extension of the opposite side).

<note>In the figure of this problem, the segment 23 feet perpendicular to the lower base of 37 feet from a point on the upper base of 22 feet is the height of the trapezoid</note>. <note>The segment 23 feet forms a right angle (90 degrees) with the lower base of 37 feet</note>, which indicates that <note>the segment 23 feet is the vertical distance from the upper base of 22 feet to the lower base of 37 feet</note>.

Trapezoid Area Formula

Given a trapezoid with the lengths of its two parallel sides as \(a\) and \(b\), and the height between these sides as \(h\), the area of the trapezoid is given by \(\frac{(a + b) \cdot h}{2}\).

<note>The upper base of the trapezoid is 22 feet</note>, <note>the lower base is 37 feet</note>, <note>the height is 23 feet</note>, so the area of the trapezoid is (22 feet + 37 feet) * 23 feet / 2.


Question: In the figure above, two line segments meet at a point on line l. What is the value of x?

Answer: 45

1. Given that two line segments intersect on a straight line forming three angles of x°, 2x°, x°, according to the definition of a straight angle, we get x° + 2x° + x° = 180°.

2. Simplify the equation x° + 2x° + x° = 180° to 4x = 180, thus deriving x = 180 / 4.

3. Calculate 180 / 4 = 45.

4. Through the above reasoning, the final answer is x = 45.

对顶角; 邻补角; 直线

Definition of an Angle

An angle is a geometric figure formed by two rays sharing a common endpoint. This common endpoint is called the vertex of the angle, and the two rays are referred to as the sides of the angle.

The original text: <note>angle x°</note>, <note>2x°</note>, <note>x°</note> is a geometric figure composed of two rays, which have a common endpoint. This common endpoint is called the <note>vertex of the angle</note>, and the <note>rays are called the sides of the angle</note>.

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.

In the figure of this problem, <note>the ray rotates around the endpoint to form a straight line with the initial side</note>, forming a <note>straight angle</note>. According to the <note>definition of a straight angle</note>, <note>a straight angle measures 180 degrees</note>.


Question:
What is the length of the hypotenuse in the triangle above?
a. √11
b. 8
c. √65
d. 11
e. 65

Answer: \sqrt{65}

1. Given that the triangle is a right triangle with legs of 7 and 4, according to the Pythagorean theorem, let the two legs of the right triangle be a and b, and the hypotenuse be c, then the formula is: ##c?= a? + b?##.

2. Substitute the given conditions into the Pythagorean theorem formula: a = 7, b = 4, then ##c? = 7? + 4?##.

3. Calculate the terms in the formula: ##7? = 49##, ##4? = 16##, so ##c? = 49 + 16 = 65##.

4. Find the value of c: c = √65.

5. Through the above reasoning, the final answer is √65.

Definition of Right Triangle

<note>One angle of the triangle is a right angle (90 degrees)</note>, therefore the triangle is a right triangle. <note>The segments with lengths 7 and 4 are the legs</note>, <note>the hypotenuse is the segment with length √65</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.

<note>In the diagram of this problem</note>, in a right-angled triangle, <note>the lengths of the legs are 7 and 4, and the hypotenuse is c</note>, so according to the Pythagorean Theorem, <note>##c? = 7? + 4?##= 49 + 16 = 65</note>, therefore <note>c = √65</note>.


Question: If x = 32 and r = 18, what is the length of the arc shown in the figure above?

Answer: 16*\pi/5

Given the conditions angle x = 32 and radius r = 18, ##let the center of the circle be O, the upper radius intersects the circle at point A, and the lower radius intersects the circle at point B. According to the formula for the arc length of a sector, the arc length L = θ * r (θ is expressed in radians)## start calculating the arc length.

##Based on the conversion formula between degrees and radians, convert 32 degrees to radians as 32 * π / 180 = 16π / 90, and substitute it into the formula together with the radius r = 18: L = (16π / 90) * 18##.

####

##3##. After the above calculation, the length of the arc is (16/5)π.

##4##. The final answer is 16π/5.

Formula for the Length of an Arc of a Sector

The length \( L \) of the arc of a sector is equal to the central angle \( \theta \) (measured in radians) multiplied by the radius \( r \): \( L = \theta r \).

In the figure of this problem, in the sector, <note>the central angle is x = 32 degrees</note>, <note>the radius is r = 18</note>. According to the formula for the length of an arc of a sector, <note>the arc length L is equal to the central angle θ (expressed in radians) multiplied by the radius r</note>, i.e., L = θ * r. Converting 32 degrees to radians is <note>32 * π / 180 = 16π / 90</note>, substituting into the formula <note>L = (16π / 90) * 18</note>.

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, in the circle, <note>point O is the center of the circle, and the radius is 18</note>. <note>All points in the figure that are at a distance of 18 from point O are on the circle</note>.

Definition of Arc

An arc is a curve on a circle that is defined by two distinct points on the circle.

<note>There are two points A and B on the circle</note>, arc AB is a segment of the curve connecting these two points. According to the definition of an arc, <note>arc AB is a segment of the curve between the two points A and B on the circle</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.

Original text: <note>In circle O, points A and B are two points on the circle, and the center of the circle is point O</note>. <note>The angle ∠AOB formed by the lines OA and OB</note> is called the central angle.

Definition of Radius

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

<note>In circle O, point O is the center of the circle</note>, <note>point A is any point on the circle</note>, <note>line segment OA is the line segment from the center of the circle to any point on the circle</note>, therefore <note>line segment OA is the radius of the circle</note>. <note>Point B is any point on the circle</note>, <note>line segment OB is the line segment from the center of the circle to any point on the circle</note>, therefore <note>line segment OB is the radius of the circle</note>.

Definition of Sector

A sector is a figure formed by two radii of a circle and the arc enclosed between them.

In the figure of this problem, <note>sector OAB</note>, <note>radius OA and radius OB are two radii of the circle, and arc AB is the arc enclosed by these two radii</note>, so according to the definition of sector, the figure formed by these two radii and <note>the arc AB enclosed by them is a sector</note>.

Formula for Conversion between Degrees and Radians

Radians = Degrees × (π/180), Degrees = Radians × (180/π)

In the figure of this problem, circle <note>O, ∠AOB=32°</note>, according to the <note>formula for conversion between degrees and radians</note>, converted to <note>radians = degrees * (π/180), i.e., 32° * (π/180)</note>