Question: A square is inscribed in a circle with a radius of 6 inches. Find the side length of the square.

Answer: 6 \sqrt 2

1. Given that square ABCD is inscribed in circle O, according to the definition of diameter, the diameter of the circle is equal to twice the radius, so the diameter of the circle is 12 inches.

2. According to the definition of a square, ∠BAD=∠ABC=∠BCD=∠ADC=90°, AB=BC=CD=DA, connect the diagonal vertices B and D to form diagonal BD. According to (Corollary 2 of the Inscribed Angle Theorem), the inscribed angle subtended by the diameter is a right angle, so BD is the diameter, thus BD=12.

3. Because AB=DA and ∠BAD=90°, according to the definition of an isosceles right triangle, △ABD is an isosceles right triangle. According to the Pythagorean theorem, we have AB² + DA² = BD², that is AB² + AB² = BD²; 2AB² = BD².

4. Substituting the values, we get 2AB² = 12².

5. After calculation, we obtain 2AB² = 144, simplifying to AB² = 72.

6. Further calculation gives AB = √72 = 6√2. Therefore, the side length of the square is 6√2 inches.

7. Finally, we conclude that the side length of the square is 6√2 inches.

正方形; 圆; 直角三角形; 线段

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 <note>circle O</note>, <note>the vertex A of angle BAD is on the circumference</note>, <note>the two sides of angle BAD intersect circle O at points B and D respectively</note>. Therefore, <note>angle BAD is an inscribed angle##</note>.

Definition of Diameter

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

<note>BD is the diameter</note>, connecting <note>the center O and points B and D on the circumference</note>, with a length of <note>2 times the radius, i.e., BD = 12</note>.

Definition of Square

A quadrilateral is a square if and only if all four sides are congruent (≅) and all four interior angles are right angles (90°).

<note>Because quadrilateral ABCD is a square</note>, therefore <note>∠BAD=∠ABC=∠BCD=∠ADC=90°</note>, <note>AB=BC=CD=DA</note>.

Definition of Diagonal

A diagonal is a line segment connecting one vertex of a polygon to another vertex that is not adjacent to it.

In the <note>square ABCD</note>, the vertices are A, B, C, and D respectively, <note>the diagonal is the line segment connecting vertex B and the non-adjacent vertex D</note>. Therefore, <note>the line segment BD is the diagonal of square ABCD</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>Triangle ABD is an isosceles right triangle</note>, where <note>angle BAD is a right angle (90 degrees)</note>, and <note>sides AB and DA are equal-length legs</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 isosceles right triangle ABD</note>, <note>angle BAD is a right angle (90 degrees)</note>, sides AB and DA are the legs and are equal, side <note>BD is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>BD² = AB² + DA²</note>, that is, <note>BD² = AB² + AB² = 2AB²</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>The inscribed angle BAD is 90 degrees</note>, so <note>the chord BD it subtends is the diameter</note>##.


Question: As shown in the figure, AB and CD are two chords of ⊙O. Connect AD and BC. If ∠BCD=50°, then the degree of ∠BAD is ()

Answer: 50°

2. According to the corollary of the Inscribed Angle Theorem, in a circle, any two inscribed angles that correspond to the same arc are equal. ∠BAD and ∠BCD correspond to the same arc BD, therefore ∠BAD=∠BCD.

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

圆; 弦; 圆周角; 内错角; 圆内接四边形

Definition of Inscribed Angle

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

In a circle, any two inscribed angles that subtend the same arc are equal.

In <note>circle O</note>, the inscribed angles <note>∠BAD</note> and <note>∠BCD</note> corresponding to <note>arc BD</note> are equal. According to Corollary 1 of the Inscribed Angle Theorem, the inscribed angles <note>∠BAD</note> and <note>∠BCD</note> corresponding to the same <note>arc BD</note> are equal, i.e., <note>∠BAD = ∠BCD</note>.


Question: Solve for x so that the quadrilateral is a parallelogram.

Answer: 34

1. ##Let the quadrilateral be parallelogram ABCD (clockwise from the top left corner). According to the properties and definition of parallelogram, opposite angles are equal, opposite sides are equal, diagonals bisect each other, and opposite sides are parallel. Therefore, ∠B=∠D, i.e., (3y-4)°=(4x-8)°##.

2. ##According to the parallel axiom 2 and the definition of alternate interior angles, since AB∥CD and AB and CD are intersected by diagonal AC, ∠BAC=∠ACD (alternate interior angles are equal), i.e., (x-12)°=(1/2)y°##.

3. ##Solve the equations: x-12=(1/2)y, solving gives: y=2x-24. Substitute y=2x-24 into the equation 3y-4=4x-8, we get: 6x-76=4x-8. Moving terms to combine, we get: 6x-4x=-8+76, finally solving gives: x=34##.

####

##4##. Through the above reasoning, the value of x is 34 to make the quadrilateral a parallelogram.

平行四边形; 同旁内角; 内错角; 平行线

Definition of Alternate Interior Angles

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

In the figure of this problem, <note>two parallel lines AB and CD</note> are intersected by <note>a line AC</note>, where <note>angle BAC and angle ACD are between the two parallel lines and on opposite sides of the intersecting line AC</note>. Therefore, <note>angle BAC and angle ACD are alternate interior angles</note>. Alternate interior angles are equal, i.e., <note>angle BAC is equal to angle ACD</note>.

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>A quadrilateral is a parallelogram</note>, <note>sides AB and CD are parallel and equal</note>, <note>sides AD and BC are parallel and equal</note>. According to the properties of a parallelogram, <note>opposite angles ∠B and ∠D are equal</note>, and <note>opposite angles ∠DAB and ∠BCD are equal</note>.

Properties of Parallelogram Theorem

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

In the figure of this problem, in the parallelogram, <note>the opposite angles ∠DAB and ∠BCD are equal, and the opposite angles ∠B and ∠D</note> are equal; sides AB and CD are equal, sides BC and AD are equal; the diagonals AC and BD bisect each other, <note>let the two diagonals intersect at point E, point E</note> divides diagonal AC into two equal segments AE and EC, and divides diagonal BD into two equal segments BE and ED.

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 diagram of this problem, two parallel lines AB and CD are intersected by a third line AC</note>, forming the following geometric relationships: 1. Corresponding angles: none. 2. <note>Alternate interior angles: angle BAC and angle ACD are equal</note>. 3. Consecutive interior angles: none. These relationships illustrate that when two parallel lines are intersected by a third line, <note>alternate interior angles are equal</note>.


Question: Find x.

Answer: 4 \sqrt { 6 }

1. Let the three vertices of the triangle with the right angle symbol in the figure be A, B, C.

2. Given ∠ABC = 90°, according to the definition of a right triangle, triangle ABC is a right triangle.

3. According to the Pythagorean theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two right-angle sides. The formula is: c^2 = a^2 + b^2.

4. In the problem, the x to be solved is the right-angle side AB of the right triangle, given that the right-angle side BC is 10 and the hypotenuse AC is 14.

5. Substitute the side lengths provided in the figure into the Pythagorean theorem formula: 14^2 = 10^2 + x^2.

7. Then take the square root of the result: x = √96.

9. Finally, after calculation, the answer is: x = 4√6.

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

Pythagorean Theorem

In the right triangle ABC, the legs are 10 and <note>4√6</note>, and the hypotenuse is <note>x</note>. According to the Pythagorean Theorem, <note>14?</note> = 10? + <note>x?</note>. Calculated as: <note>√x = 196-100</note>, finally <note>x = 4√6</note>.


Question: Find x.

Answer: 30

1. According to the figure, T is a point on the circumference, RT is the diameter. According to the definition of the inscribed angle, ∠RST is an inscribed angle. According to (Corollary 2 of the Inscribed Angle Theorem), the inscribed angle subtended by the diameter is a right angle, thus ∠RST=90°.

2. According to the sum of the interior angles of a triangle, x+2x+∠RST=180°, rearranging gives 3x=180°-90°=90°, thus x=30°.

3. Through the above reasoning, the final answer is 30°.

Definition of Circle

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

In the figure of this problem, <note>the center of the circle is the center point of the circle</note>, all points are at an equal distance to <note>this center point</note>, including <note>points R, S, T</note>. <note>The center of the circle is the blue point in the figure</note>, <note>the radius is RT</note>, all points in the figure that are at a distance equal to RT from <note>this center point</note> are on <note>this circle</note>.

Definition of Diameter

<note>RT is the diameter</note>, connecting the center of the circle and the points <note>R and T</note> on the circumference, with a length equal to 2 times the radius <note>RT = 2r</note>.

Definition of Inscribed Angle

<note>The vertex S of angle RST is on the circumference</note>, and the two sides of angle RST intersect the circle at <note>point R and point T</note>. Therefore, <note>angle RST is an inscribed angle</note>.

The angle subtended by the diameter <note>RT of the circle</note> is a right angle (<note>∠RST is 90 degrees</note>).

Triangle Angle Sum Theorem

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

<note>In triangle RST, ∠SRT, ∠RST, and ∠STR are the three interior angles of triangle RST</note>, according to the Triangle Angle Sum Theorem, <note>∠SRT + ∠RST + ∠STR = 180°</note>.


Question: P Q R S is a rhombus inscribed in a circle. Find m \widehat S P.

Answer: 90

1. Given PQRS is a rhombus, and it is inscribed in a circle, ##the four vertices of the rhombus are on the circle##.

2. In the circle, ##according to the properties of the rhombus's diagonals##, the diagonals of the rhombus bisect each other and are perpendicular to each other.

3. ##Let the intersection point of the diagonals PR and QS of the rhombus be point O. Since PR⊥QS, ∠POS is a right angle, i.e., ∠POS=90°##.

4. ##Since circle O is the circumscribed circle of rhombus PQRS, the intersection point of the diagonals of the rhombus is the center of circle O. The central angle corresponding to arc SP is ∠POS, so arc SP=90°##.

##5##. Through the above reasoning, the final answer is ##arc SP= 90°##.

菱形; 圆; 圆内接四边形; 圆周角

Definition of Circle

<note>In circle O, point O is the center of the circle</note>. In the figure, <note>all points that are at a distance equal to the radius from point O lie on circle O</note>.

Definition of Rhombus

A quadrilateral is a rhombus if and only if all four sides are of equal length, and its diagonals are perpendicular bisectors of each other.

<note>In quadrilateral PQRS, all sides PQ, QR, RS, and SP are equal</note>, so quadrilateral PQRS is a rhombus. Additionally, <note>the diagonals PR and QS of quadrilateral PQRS are perpendicular bisectors of each other</note>, that is, <note>the diagonals PR and QS intersect at point O</note>, and <note>angle POQ is a right angle (90 degrees)</note>, and <note>PO=OR and QO=OS</note>.

Properties of the Diagonals of a Rhombus

In a rhombus, the diagonals bisect each other and are perpendicular to each other.

In <note>rhombus PQRS</note>, <note>diagonals PR and QS bisect each other and are perpendicular to each other</note>. Specifically, <note>point O is the intersection of diagonals PR and QS, and PO = OR and QO = OS</note>. Additionally, <note>angle POS and angle QOR are both right angles (90 degrees)</note>, so <note>diagonals PR and QS are perpendicular to each other</note>.

Properties of Central Angles

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

In the figure of this problem, it is known that the central angle ∠POS corresponds to the arc SP. According to the properties of central angles, the degree of the central angle is equal to the degree of the corresponding arc, i.e., the degree of ∠POS = the degree of arc SP.


Question: Circle A has diameters DF and PG. If DF = 10, find DA.

Answer: 5

1. ##Given DF is the diameter of circle A, and DF = 10. According to the definition of radius, DA is the radius of circle A.##

2. The diameter of circle A, DF = 10, then according to the definition of diameter, the radius of the circle is half of the diameter, i.e., the radius of the circle is 10 / 2 = 5.

####

##3##. Through the above reasoning, the final value of DA is 5.

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, DF is the diameter</note>, connecting <note>the center of the circle A</note> and <note>points D and F on the circumference</note>, with a length of <note>2 times the radius</note>, that is, <note>DF = 2 * DA</note>.

Definition of Circle

In <note>Circle A</note>, <note>point A is the center</note>, and <note>the radius is DA</note>. All points in the figure that are <note>at a distance of 10/2 = 5 from point A</note> lie on <note>Circle A</note>.

Definition of Radius

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

<note>In circle A, point A is the center of the circle</note>, <note>point D is any point on the circle</note>, <note>line segment AD is the line segment from the center to any point on the circle</note>, therefore <note>line segment AD is the radius of the circle</note>.


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

Answer: 76

1. Given conditions: This is a parallelogram, where one side length is 21 ft, and another side length is 17 ft. According to the properties theorem of parallelograms, the two pairs of opposite sides of a parallelogram are equal.

2. From the previous step, we can deduce that the four sides of the parallelogram are 21 ft and 17 ft each having two sides. According to the perimeter of the parallelogram, the formula for the perimeter of a parallelogram is: Perimeter = 2*a + 2*b, where a and b are the lengths of the adjacent sides of the parallelogram.

3. Substitute the known side lengths, the perimeter of the parallelogram is calculated as: Perimeter = 2*21 + 2*17 = 42 + 34 = 76 ft.

4. Through the above reasoning, the final answer is 76 ft.

Definition of Parallelogram

<note>The two pairs of opposite sides of a parallelogram are 21 feet and 17 feet respectively</note>, the two opposite sides of 21 feet are <note>parallel and equal</note>, and the two opposite sides of 17 feet are <note>parallel and equal</note>.

Properties of Parallelogram Theorem

<note>In the figure of this problem</note>, the two pairs of <note>opposite sides of the parallelogram are equal</note>, that is, one of the <note>sides</note> is <note>21 feet</note> long, and the other <note>side</note> is <note>17 feet</note> long. According to the theorem, the lengths of the <note>opposite sides corresponding to these two sides</note> are also <note>21 feet</note> and <note>17 feet</note>.

Perimeter of a Parallelogram

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

<note>The lengths of the two pairs of opposite sides of the parallelogram are 21 feet and 17 feet</note>. According to <note>the formula for the perimeter of a parallelogram</note>, the perimeter of the parallelogram is equal to twice the sum of its two pairs of adjacent sides, which is <note>Perimeter P = 2(21 + 17) = 76 feet</note>.


Question: Use parallelogram ABCD to find AD.

Answer: 18

1. From the figure, it can be obtained that in the parallelogram ABCD, side BC = 18.

2. According to the definition of a parallelogram, opposite sides of a parallelogram are equal, that is AD = BC.

3. From the above property, it can be concluded that AD = 18.

Definition of Parallelogram

<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>.


Question: As shown in the figure, in the cyclic quadrilateral ABCD, the central angle ∠1=100°, then the inscribed angle ∠ABC equals ()

Answer: 130°

2. According to the inscribed angle theorem, the inscribed angle ∠ADC is equal to half of the corresponding central angle ∠AOC, thus ∠ADC=1/2∠AOC=50°.

3. Since ABCD is a cyclic quadrilateral, according to the corollary of the inscribed angle theorem (cyclic quadrilateral opposite angle theorem), ∠ABC+∠ADC=180°.

4. Given ∠ADC=50°, then ∠ABC=180°-50°=130°.

5. Therefore, the final conclusion is ∠ABC=130°.

圆内接四边形; 圆心角; 圆周角; 弦; 圆

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>Point A and Point D</note> are two points on the circle, <note>the center of the circle is Point O</note>. The angle formed by the lines <note>OA and OD</note>, <note>∠AOD</note>, is called the central angle.

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, in circle O, points A, B, C, and D are on the circle, <note>the central angle corresponding to arc AD is ∠AOC, and the inscribed angle is ∠ADC</note>. According to the Inscribed Angle Theorem, <note>∠ADC is equal to half of the central angle ∠AOC corresponding to arc AD, that is, ∠ADC = 1/2 ∠AOC</note>.

Corollary 3 of the Inscribed Angle Theorem: Diagonal Supplementary Theorem for Cyclic Quadrilateral

In a cyclic quadrilateral (a quadrilateral inscribed in a circle), the sum of each pair of opposite angles is equal to 180°. Additionally, any exterior angle is equal to its interior opposite angle.

<note>Quadrilateral ABCD is a cyclic quadrilateral, with vertices A, B, C, and D all on the circle</note>. According to the Diagonal Supplementary Theorem for Cyclic Quadrilateral, <note>the sum of each pair of opposite angles of quadrilateral ABCD is equal to 180°</note>. Specifically, <note>∠ABC + ∠ADC = 180°; ∠BAD + ∠BCD = 180°</note>.

Cyclic Quadrilateral

A quadrilateral is called a cyclic quadrilateral if and only if all four vertices lie on the circumference of a single circle. This circle is referred to as the circumcircle of the quadrilateral.

In the figure of this problem, <note>the four vertices A, B, C, and D of quadrilateral ABCD are all on the same circle</note>. This circle is called <note>the circumcircle of quadrilateral ABCD</note>. Therefore, <note>quadrilateral ABCD is a cyclic quadrilateral</note>. According to the properties of cyclic quadrilaterals, it can be concluded that <note>the sum of opposite angles is equal to 180 degrees</note>, i.e., <note>angle ABC + angle ADC = 180 degrees</note>, <note>angle BAD + angle BCD = 180 degrees</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 original text: <note>In circle O, the vertex of angle ADB (point D) is on the circumference</note>, <note>the two sides of angle ADC intersect circle O at points A and C, respectively</note>. Therefore, <note>angle ADC is an inscribed angle</note>.


Question: M N \parallel B C. Solve for x.

Answer: 9

1. Given the condition MN ∥ BC, according to the parallel axiom 2 of parallel lines, corresponding angles are equal, we can obtain ∠AMN = ∠ABC, ∠ANM = ∠ACB.

2. Since ∠AMN = ∠ABC, ∠ANM = ∠ACB, according to the AA similarity theorem, △AMN is similar to △ABC.

3. According to the definition of similar triangles, AM/AB = AN/AC.

4. As shown in the figure, in △AMN and △ABC, it is known that AM = 4x - 6, AB = 24+4x - 6, AN = 3x - 2, AC = 20+3x - 2, substituting into AM/AB = AN/AC, we get 4x - 6/(24+4x - 6) = 3x - 2/(20+3x - 2).

5. Simplifying the above equation, we get 4x - 6/18+4x = 3x - 2/18+3x.

7. Expanding and simplifying the equation, we get 12x?+54x-108 = 12x?+46x-36.

8. Moving all x terms to one side and constant terms to the other side, we get 12x?+54x-12x?-46x = 108-36.

10. Solving this equation, we get x = 9.

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

Definition of Parallel Lines

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

<note>Line MN and line BC</note> are located in the same plane and do not intersect, so according to the definition of parallel lines, <note>line MN and line BC</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>In the figure of this problem, triangle AMN and triangle ABC are similar triangles</note>. According to the definition of similar triangles, we have: <note>∠MAN = ∠BAC</note>, <note>∠ANM = ∠ACB</note>, <note>∠AMN = ∠ABC</note>; <note>AM/AB = AN/AC = MN/BC</note>.

Similarity Theorem for Triangles (AA)

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

<note>In the figure of this problem, in triangles AMN and ABC, if angle AMN is equal to angle ABC, and angle ANM is equal to angle ACB, then triangle AMN is similar to triangle ABC.</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>Two parallel lines MN and BC</note> are intersected by a third line <note>AB</note>, <note>two parallel lines MN and BC</note> are intersected by a third line <note>AC</note>, forming the following geometric relationships: <note>corresponding angles: angle AMN and angle ABC are equal, angle ANM and angle ACB are equal.</note>

Definition of Corresponding Angles

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

<note>Two parallel lines MN and BC are intersected by a line AB</note>, where <note>angle AMN and angle ABC are on the same side of the intersecting line AB</note>, and are on the <note>same side of the two intersected lines MN and BC</note>, thus <note>angle AMN and angle ABC are corresponding angles</note>; <note>Two parallel lines MN and BC are intersected by a line AC</note>, where <note>angle ANM and angle ACB are on the same side of the intersecting line AC</note>, and are on the <note>same side of the two intersected lines MN and BC</note>, thus <note>angle ANM and angle ACB are corresponding angles</note>. <note>Corresponding angles are equal</note>, that is, <note>angle AMN is equal to angle ABC</note>, <note>angle ANM is equal to angle ACB</note>.


Question: For trapezoid T R S V, M and N are the midpoints of the two legs. If T R = 32 and M N = 25, find V S.

Answer: 18

1. Given that T R S V is a trapezoid, M and N are the midpoints of the legs of the trapezoid, i.e., the midpoints of T V and R S, T R = 32, M N = 25.

2. According to the trapezoid midsegment theorem, the midsegment of a trapezoid is parallel to the two bases and its length is half the sum of the lengths of the two bases.

3. Let V S be x, then according to the theorem, M N = (T R + V S) / 2.

4. Substitute the given conditions into the above equation to get 25 = (32 + x) / 2.

5. Solve the equation 25 = (32 + x) / 2, first multiply both sides of the equation by 2 to get 50 = 32 + x.

6. Solve for x to get x = 50 - 32.

7. The result of the calculation is x = 18.

8. Therefore, through the above reasoning, the final answer is 18.

Definition of Trapezoid

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

In the quadrilateral <note>TRSV</note>, <note>sides TR and VS are parallel</note>, while <note>sides TV and RS are not parallel</note>. Therefore, according to the definition of a trapezoid, <note>the quadrilateral TRSV is a trapezoid</note> because it has exactly one pair of parallel sides.

Midpoint of a Line Segment

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

In the figure of this problem, <note>M is the midpoint of line segment T V</note>, that is, <note>T M = M V</note>; <note>N is the midpoint of line segment R S</note>, that is, <note>R N = N S</note>.

Median Line Theorem of Trapezoid

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

<note>In trapezoid T R S V, sides T R and V S are the two bases of the trapezoid</note>, <note>points M and N are the midpoints of the legs T V and R S of the trapezoid</note>, <note>segment M N is the median line connecting the midpoints of the legs</note>. According to the Median Line Theorem of Trapezoid, <note>segment M N is parallel to sides T R and V S</note>, and <note>the length of segment M N is equal to half the sum of the lengths of sides T R and V S</note>, that is, <note>M N = (T R + V S) / 2</note>.


Question: As shown in the figure, in △ABC, it is known that ∠A=80°, ∠B=60°, DE∥BC, then the size of ∠CED is ()

Answer: 140°

1. First, it is known that in triangle ABC, angle BAC is equal to 80 degrees, and angle ABC is equal to 60 degrees.

2. According to the triangle angle sum theorem, the sum of the three interior angles of triangle ABC is 180 degrees, ##angle BAC + angle ABC + angle ACB = 180 degrees##, that is, 80 degrees + 60 degrees + angle ACB = 180 degrees. Therefore, angle ACB is equal to 180 degrees minus angle BAC minus angle ABC, that is, ##angle ACB = 180 degrees - 80 degrees - 60 degrees = 40 degrees##.

3. Since segment DE is parallel to segment BC, according to the parallel line axiom 2, the interior angles on the same side are supplementary, angle CED plus angle ACB equals 180 degrees.

4. Therefore, angle CED is equal to 180 degrees minus angle ACB, that is, 180 degrees minus 40 degrees equals 140 degrees.

5. Through the above reasoning, the final answer is 140 degrees.

平行线; 同位角; 内错角; 普通三角形

Triangle Angle Sum Theorem

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

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

Definition of Consecutive Interior Angles

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

In the diagram of this problem, <note>two lines DE and BC are intersected by a third line AC</note>, <note>two angles CED and ACB are on the same side of the intersecting line AC</note>, <note>and within the intersected lines DE and BC</note>, so <note>angles CED and ACB are consecutive interior angles</note>. <note>Consecutive interior angles CED and ACB are supplementary</note>, that is, <note>angle CED + angle ACB = 180 degrees</note>.

Parallel Postulate 2 of Parallel Lines

<note>Two parallel lines DE and BC</note> are intersected by <note>a third line AC</note>, forming the following geometric relationships: 1. Corresponding angles: <note>angle ADE and angle ABC are equal</note>. ###### 2. Consecutive interior angles: <note>angle CED and angle ACB are supplementary, that is, angle CED + angle ACB = 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.####


Question: In triangle CDF, K is the centroid and DK = 16. Find CD.

Answer: 18

1. Given DG = 9, K is the centroid of triangle CDF. According to the centroid theorem, the centroid is the intersection point of the three medians of the triangle. Therefore, FG is a median of triangle CDF with F as the vertex, drawn to side CD.

2. According to the definition of a triangle's median, FG bisects CD at point G, i.e., DG = GC.

3. Therefore, DG = GC = 9, CD = DG + GC = 9 + 9 = 18.

4. Through the above reasoning, the final answer is 18.

普通三角形; 线段; 中点

Centroid Theorem

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

In triangle CDF, point K is the centroid of the triangle. The three medians of the triangle are line segment CJ, line segment DH, and line segment FG, and they intersect at point K.

Definition of Median of a Triangle

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

<note>Vertex F is a vertex of the triangle, and the opposite side CD is the side opposite to vertex F</note>. Point G is <note>the midpoint of side CD, and segment FG is the segment from vertex F to the midpoint G of the opposite side CD</note>, <note>point G bisects CD, that is, DG = GC</note>.


Question: As shown in the figure, △ABC is an inscribed triangle of ⊙O, ∠OAB=35°, then the degree of ∠ACB is ()

Answer: 55°

1. Given ∠OAB=35°, since OA and OB are both radii, OA=OB. According to the properties of an isosceles triangle, ∠OBA=∠OAB=35°.

2. In △OAB, the sum of the interior angles of a triangle is 180°. According to the interior angle sum theorem, ∠OAB + ∠OBA + ∠AOB=180°, we get ∠AOB=180° - 35° - 35° = 110°.

3. According to the inscribed angle theorem, the inscribed angle is equal to half of the central angle that subtends the same arc, we get ∠ACB=1/2 ∠AOB.

4. Substituting the values, we get ∠ACB=1/2 * 110° = 55°.

5. Through the above reasoning, we finally get the answer: ∠ACB = 55°.

圆; 圆周角; 三角形的外角; 普通三角形

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 OAB, side OA and side OB are equal</note>. Therefore, according to the properties of an isosceles triangle, the angles opposite the equal sides are equal, i.e., <note>angle OAB = angle OBA = 35°</note>.

Definition of Inscribed Angle

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

Triangle Angle Sum Theorem

<note>In triangle OAB, ∠OAB, ∠OBA, and ∠AOB are the three interior angles of triangle OAB</note>, according to the Triangle Angle Sum Theorem, <note>∠OAB + ∠OBA + ∠AOB = 180°</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, in circle O, <note>points A, B, and C are on the circle</note>, <note>the central angles corresponding to arc AC and arc CB are ∠AOB = 110°</note>, and the inscribed angle is <note>∠ACB</note>. According to the Inscribed Angle Theorem, <note>∠ACB is equal to half of the central angle ∠AOB corresponding to arc AC</note>, that is, <note>∠ACB = 1/2 ∠AOB</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, triangle <note>OAB</note> has sides <note>OA</note> and <note>OB</note> equal, so triangle <note>OAB</note> is an isosceles triangle.

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 circle O, point A and point 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>.


Question: Find GH in the kite.

Answer: \sqrt { 369 }

1. Given the kite FGHI, let the diagonals FH and GJ intersect at point O. From the figure, we know GO=12 and HO=15. According to the properties of a kite, FH⊥GJ, thus ∠GOH=90°.

2. Since ∠GOH=90°, according to the definition of a right triangle, △GOH is a right triangle.

3. In Rt△GOH, according to the Pythagorean theorem, GH? = GO? + HO?.

4. Substituting the values into the equation, we get GH? = 12? + 15? = 144 + 225 = 369, thus GH = √369.

5. After the above calculations, the final answer is √369.

菱形; 等腰三角形; 垂直平分线

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, <note>triangle GOH</note> has a <note>right angle at angle GOH (90 degrees)</note>, so <note>triangle GOH is a right triangle</note>. <note>Sides GO and HO are the legs</note>, and <note>side GH is the hypotenuse</note>.

Pythagorean Theorem

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

<note>In the right triangle GOH, angle GOH is a right angle (90 degrees)</note>, side <note>GO and HO are the legs</note>, side <note>GH is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>GH² = GO² + HO²</note>.

Definition of Kite

A quadrilateral is a kite if and only if it has one diagonal that acts as the axis of symmetry for the shape.

<note>Quadrilateral FGHJ</note> has <note>a diagonal FH as the axis of symmetry</note>, causing the quadrilateral to fold completely along this line, thus the quadrilateral is a kite.

Properties of a Kite

The two diagonals of a kite are perpendicular to each other, and only one of the diagonals is bisected by the other.

<note>The diagonals FH and GJ of kite FGHJ intersect at point O</note>. According to the properties of a kite, <note>its two diagonals FH and GJ are perpendicular to each other, i.e., FH⊥GJ</note>, and only one of the diagonals <note>GJ is bisected by the other diagonal FH, i.e., GO=JO</note>.


Question: Find X W.

Answer: 12

1. Given that WZ = ZY, and XZ is perpendicular to WY, Z is the midpoint of WY, ##so according to the definition of the perpendicular bisector,## XZ is the ##perpendicular bisector## of WY.

2. Since XZ is the ##perpendicular bisector## of WY, according to the properties of the perpendicular bisector, any point on the perpendicular bisector is equidistant from the endpoints of the segment, therefore WX = XY.##

####

##3##. From the figure provided, the length of XY is given as 12.

##4## Therefore, it can be deduced that WX is also equal to 12.

##5##. Through the above reasoning, the final answer is 12.

等腰三角形; 垂直平分线; 中点; 直角三角形

Midpoint of a Line Segment

<note>The midpoint of line segment WY is point Z</note>. According to the definition of the midpoint of a line segment, <note>point Z divides line segment WY into two equal parts</note>, that is, <note>the lengths of line segment WZ and line segment ZY are equal</note>. That is, <note>WZ = ZY</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.

<note>Line XZ passes through the midpoint Z of segment WY</note>, and <note>line XZ is perpendicular to segment WY</note>. Therefore, <note>line XZ is the perpendicular bisector of segment WY</note>.

Properties of the Perpendicular Bisector

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

The perpendicular bisector of <note>segment WY</note> is <note>line XZ</note>, and <note>point X is on line XZ</note>. According to the <note>properties of the perpendicular bisector</note>, <note>the distance from point X to the endpoints W and Y of segment WY is equal</note>, that is, <note>XW = XY</note>.


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

Answer: 0.38

1. Given that ∠P = 90 degrees, triangle NPM is a right triangle, and it is known that PM = 36, PN = 15, NM = 39.

2. According to the cosine function, cos(∠N) = NP/MN.

3. Given PN = 15, NM = 39, substituting the values we get: cos(∠N) = 15/39 ≈ 0.384. According to the problem requirements, round to two decimal places, so cos(∠N) ≈ 0.38.

4. Through the above reasoning, the cosine value of ∠PNM is 0.38.

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 PNM is a geometric figure composed of three non-collinear points P, N, M and their connecting line segments PN, PM, MN</note>.<note>Points P, N, M are the three vertices of the triangle</note>,<note>Line segments PN, PM, MN are the three sides of the triangle</note>.

Definition of Right Triangle

In <note>triangle PNM</note>, <note>angle NPM is a right angle (90 degrees)</note>, therefore <note>triangle PNM is a right triangle</note>. <note>Side PN and side PM are the legs</note>, <note>side MN 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.

<note>In the right triangle NPM</note>, <note>side NP is the adjacent side of angle ∠PNM</note>, <note>side NM is the hypotenuse</note>. Therefore, according to the definition of the cosine function, <note>the cosine of angle ∠PNM is equal to the ratio of the adjacent side NP to the hypotenuse NM</note>, that is, <note>cos(∠PNM) = NP / NM</note> = 15 / 39.


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

Answer: 182.6

1. Given that ∠RST = 93° and SR = ST = 15 mm, we can start solving based on the definitions of central angle and sector area.

2. The formula for the area of a sector is: A = ##(θ/360) * π * r^2##, where θ is the central angle in degrees and r is the radius length.

3. Substituting the given radius SR = 15 mm and central angle RST = 93° into the formula, we get the sector area: A = ##(93/360) * π * 15^2##.

4. Calculate ##(93/360) * π * 15^2##, first calculate (93/360) ≈ 0.2583.

5. Calculate ##π * 15^2##, first calculate ##15^2## = 225, then π * 225 = 706.8583 (π is approximately 3.1416).

6. Finally, calculate 0.2583 * 706.8583 ≈ 182.6137.

7. Round 182.6137 to the nearest tenth, getting 182.6.

8. Therefore, the calculated area of the shaded sector is ##182.6 mm^2##.

Definition of Sector

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

In the <note>sector RST</note>, the <note>radius SR</note> and <note>radius ST</note> are two radii of the circle, and the <note>arc RT</note> is the arc enclosed by these two radii. Therefore, according to the definition of a sector, the figure formed by these two radii and the enclosed <note>arc RT</note> is a sector.

Definition of Central Angle

In the circle, <note>points R and T are two points on the circle</note>, <note>the center of the circle is point S</note>. <note>The angle ∠RST formed by the lines SR and ST</note> is called the central angle.

Formula for the Area of a Sector

The area \( X \) of a sector can be calculated using the formula \( X = \frac{\theta}{360} \times \pi \times r^2 \), where \( \theta \) is the measure of the central angle in degrees, and \( r \) is the radius length.

Original text: <note>In the diagram of this problem, in sector RST, the central angle ∠RST has a degree measure of θ = 93°, and the lengths of radii SR and ST are r = 15 mm</note>. According to the formula for the area of a sector, the area A of the sector can be calculated using the formula <note>A = (θ/360) * π * r^2</note>, <note>where θ is the degree measure of the central angle and r is the length of the radius</note>. Therefore, <note>the area of sector RST A = (93/360) * π * 15^2</note>.

Definition of Radius

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

<note>In the figure of this problem</note>, in circle S, <note>point S is the center of the circle</note>, <note>point T is any point on the circle</note>, <note>line segment ST is the line segment from the center to any point on the circle</note>, therefore <note>line segment ST is the radius of the circle</note>.


Question: Find x such that G J \parallel F K. G H = x + 3.5, H J = x - 8.5, F H = 21, H K = 7.

Answer: 14.5

2. According to the parallel postulate 2 and the definition of corresponding angles, ∠HGJ = ∠HFK. According to the similarity theorem (AA), ∠HGJ = ∠HFK, ∠GHJ = ∠FHK, therefore triangle GHJ is similar to triangle FHK.

3. According to the definition of similar triangles, GH/FH = HJ/HK.

4. Substitute the given conditions, (x + 3.5) / 21 = (x - 8.5) / 7.

5. Cross-multiply to get 7(x + 3.5) = 21(x - 8.5).

6. Expand the parentheses to get 7x + 24.5 = 21x - 178.5.

7. Rearrange the equation, moving terms to get 21x - 7x = 24.5 + 178.5.

8. Simplify the equation to get 14x = 203.

11. Through the above reasoning, the final answer is x = 14.5.

Definition of Parallel Lines

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

In the figure of this problem, <note>line GJ</note> and <note>line FK</note> lie in the same plane and do not intersect, so according to the definition of parallel lines, <note>line GJ and line FK are parallel lines</note>.

Definition of Similar Triangles

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

<note>Triangles GHJ and FHK are similar triangles</note>. According to the definition of similar triangles: <note>∠GHJ = ∠FHK</note>, <note>∠HGJ = ∠HFK</note>, <note>∠GJH = ∠FKH</note>; <note>GH/FH = HJ/HK = GJ/FK</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>Two parallel lines GJ and FK</note> are intersected by a third <note>line HF</note>, forming the following geometric relationships: 1. <note>Corresponding angles: angle HGJ and angle HFK</note> are equal. These relationships illustrate that when <note>two parallel lines are intersected by a third line</note>, <note>corresponding angles are equal</note>, <note>alternate interior angles are equal</note>, <note>same-side interior angles are supplementary</note>.

Definition of Corresponding Angles

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

<note>Two parallel lines GJ and FK</note> are intersected by a <note>line HF</note>, where <note>angle HGJ and angle HFK are on the same side of the intersecting line HF</note>, <note>on the same side of the two intersected lines GJ and FK</note>. Therefore, <note>angle HGJ and angle HFK are corresponding angles</note>. Corresponding angles are equal, that is, <note>angle HGJ is equal to angle HFK</note>.

Similarity Theorem for Triangles (AA)

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

In the figure of this problem, in triangle <note>HGJ</note> and triangle <note>HFK</note>, if <note>angle HGJ is equal to angle HFK</note>, and <note>angle GHJ is equal to angle FHK</note>, then <note>triangle GHJ is similar to triangle HFK</note>.


Question: Find x. Assume that the segments that appear to be tangent are tangent. If necessary, round to the nearest tenth.

Answer: 10

2. According to the tangent length theorem, AB = AC, thus 5x - 8 = 72 - 3x.

3. Solving the equation 5x - 8 = 72 - 3x, add 3x to get 8x - 8 = 72.

4. Add 8 to both sides of the equation to get 8x = 80.

5. Divide both sides of the equation by 8 to get x = 10.

6. Through the above reasoning, the final value of x is 10.

切线; 普通三角形; 圆

Tangent-Segment Theorem

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

In the figure of this problem, the <note>two tangents</note> drawn from <note>a point outside the circle</note> are 5x - 8 and 72 - 3x respectively, <note>their tangent lengths are equal</note>, i.e., 5x - 8 = 72 - 3x. The line connecting the center of the circle y and this point <note>bisects the angle between the two tangents</note>.

Definition of Tangent to a Circle

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

<note>The circle and line AB have only one common point B, this common point is called the point of tangency</note>. Therefore, <note>line AB is the tangent to the circle</note>. <note>The circle and line AC have only one common point C, this common point is called the point of tangency</note>. Therefore, <note>line AC is the tangent to the circle</note>.


Question: In \odot O, E C and A B are diameters, and \angle B O D \cong \angle D O E \cong \angle E O F \cong \angle F O A. Find m \widehat A E.

Answer: 90

1. Given that AB is the diameter of circle O, according to the definition of a straight angle, ∠AOB = 180°, that is, ∠BOD + ∠DOE + ∠EOF + ∠FOA = 180°.

2. According to the problem, ∠BOD = ∠DOE = ∠EOF = ∠FOA, so 180°/4=45° is the degree of each angle in the semicircle.

3. According to the above steps, we can conclude ∠AOE=∠AOF+∠FOE=45°+45°=90°.

4. According to the property of central angles, the degree of a central angle is equal to the degree of its corresponding arc. ∠AOE is a central angle = 90° so arc AE = 90°.

5. Through the above reasoning, we obtain that arc AE is a right angle of 90°.

圆; 圆心角; 弧; 弦

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>EC and AB are diameters</note>, connecting the center O and the points <note>E, C, and A, B</note> on the circumference, with a length of <note>2 times the radius</note>, that is, <note>EC = 2r, AB = 2r</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.

In the figure of this problem, <note>in circle O</note>, <note>points B and C are two points on the circle</note>, <note>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>. Similarly, <note>∠DOE, ∠EOF, ∠FOA##, ∠BOD, ∠AOE, ∠AOC## are also central angles</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 E on circle O</note>, <note>arc AE is a segment of the curve connecting these two points</note>. According to the definition of an arc, <note>arc AE is a segment of the curve between points A and E on circle O</note>.

Properties of Central Angles

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

<note>The arc corresponding to the central angle ∠AOE is arc AE</note>. According to the properties of central angles, the measure of a central angle is equal to the measure of its intercepted arc, that is, <note>the measure of ∠AOE = the measure of arc AE</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.

The ray <note>OB</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>.


Question: Find m \angle Y.

Answer: 80

1. Given WXYZ is a quadrilateral, the sum of the interior angles of a quadrilateral is 360°.

2. ∠W = (1/2)a + 8, ∠X = a, ∠Y = a - 28, ∠Z = a + 2.

3. According to the theorem of the sum of the interior angles of a quadrilateral, the sum of the interior angles is 360°. Therefore, (1/2)a + 8 + a + (a - 28) + (a + 2) = 360.

4. Combining like terms, we get 3.5a - 18 = 360.

5. Solving the equation 3.5a - 18 = 360, we get 3.5a = 378, a = 108.

6. Substituting into ∠Y = a - 28, we get ∠Y = 108 - 28.

7. Calculating, we get ∠Y = 80.

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

Definition of Quadrilateral

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

<note>Quadrilateral WXYZ</note> is composed of four <note>line segments WX, XY, YZ, and ZW</note>, which are referred to as the sides of the quadrilateral. Quadrilateral WXYZ has four vertices, namely <note>point W, point X, point Y, and point Z</note>, and it has four interior angles, namely <note>angle ∠W, angle ∠X, angle ∠Y, and angle ∠Z</note>. The degrees of these angles are (1/2)a + 8, a, a - 28, a + 2 respectively.

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

In the diagram of this problem, <note>quadrilateral WXYZ</note> has <note>angles ∠W, ∠X, ∠Y, and ∠Z as the four interior angles of the quadrilateral</note>. According to the Sum of Interior Angles of a Quadrilateral Theorem, <note>the sum of these four interior angles is 360°</note>, that is, <note>∠W + ∠X + ∠Y + ∠Z = 360°</note>. Specifically, (1/2)a + 8 + a + (a - 28) + (a + 2) = 360°.


Question: As shown in the figure, points A, B, P are three points on ⊙O. If ∠AOB=40°, then the degree of ∠APB is ()

Answer: 20°

1. Given ∠AOB=40°, it is known from the problem that points A, B, and P are three points on circle O.

2. According to the inscribed angle theorem, the inscribed angle is equal to half of its corresponding central angle, i.e., ∠APB=##1/2∠AOB##.

3. Substituting the given condition ∠AOB=40°, then ∠APB=##1/2##×40°=20°.

4. Through the above reasoning, the final answer is 20°.

圆; 圆心角; 圆周角; 弦

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.

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

Definition of Central Angle

<note>In the diagram of this problem, in circle O, point A and point 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>.

Inscribed Angle Theorem

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

<note>∠APB is an inscribed angle</note>, <note>∠AOB is its corresponding central angle</note>. According to the Inscribed Angle Theorem, <note>∠APB=\frac{1}{2}∠AOB</note>.


Question: As shown in the figure, line a and line b are intersected by line c, b⊥c, and the foot of the perpendicular is point A, ∠1=70°. If line b is to be made parallel to line a, then line b should be rotated clockwise around point A by ()

Answer: 20°

1. Given that line b is perpendicular to line c, with the foot of the perpendicular being point A, according to the definition of a right angle, we have ##∠A## = 90°.

2. Given ##∠1## = 70°, and it is required that line a is parallel to line b, i.e., ##∠A and ∠1 are alternate interior angles##, according to the theorem of corresponding angles being equal for parallel lines, we have ##∠A## = 70° when a//b.

3. To make line a parallel to line b, it is necessary to rotate ##∠A to 70° and stop##.

4. Therefore, the rotation angle is ∠CAB - ∠BAC = 90° - 70° = 20°.

5. Through the above reasoning, the final answer is 20°.

平行线; 垂线; 同位角; 旋转; 直线

Definition of Alternate Interior Angles

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

<note>Two parallel lines a and b</note> are intersected by <note>a line c</note>, where <note>angle A and angle 1 are between the two parallel lines and on opposite sides of the intersecting line c</note>, thus <note>angle A and angle 1 are alternate interior angles</note>. Alternate interior angles are equal, which means <note>angle A is equal to angle 1</note>.

Parallel Postulate 2 of Parallel Lines

In the diagram of this problem, in order to make line a parallel to line b, it is necessary to make <note>alternate interior angles ∠1 = ∠A</note>. It is known that <note>∠1 = 70°</note>, so <note>∠A</note> needs to be equal to 70°.


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

Answer: \frac{4}{5}

1. Given that triangle ABC is a right triangle, where ∠ACB=90°, the length of side AC is 4, and the length of side AB is 5.

2. ##In the right triangle ABC##, we can use the ##cosine function## to calculate the cosine value of ∠BAC.

3. The ##cosine function## states that in a right triangle, the cosine value of an angle is equal to the ##ratio of the adjacent side to the hypotenuse##. For ∠BAC, its cosine value cos(∠BAC) = adjacent side (AC) / hypotenuse (AB).

4. Substitute the given values into the formula, i.e., cos(∠BAC) = AC / AB = 4 / 5.

5. After calculation, we get cos(∠BAC) = 4 / 5.

6. In summary, the cosine value of angle A is 4 / 5.

直角三角形; 余弦; 线段

Definition of Right Triangle

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

<note>In the diagram of this problem, in triangle ABC, angle ACB is a right angle (90 degrees)</note>, therefore triangle ABC is a right triangle. <note>Side AC and side BC are the legs, side AB 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, in the right triangle <note>ABC</note>, side <note>AC</note> is the adjacent side of angle <note>BAC</note>, and side <note>AB</note> is the hypotenuse. Therefore, according to the definition of the cosine function, the cosine value of angle <note>BAC</note> is equal to the ratio of the adjacent side <note>AC</note> to the hypotenuse <note>AB</note>, that is, <note>cos(∠BAC) = AC / AB</note>.


Question: Find m \angle 3.

Answer: 38

1. Given ∠TWX, ∠WXY are right angles, and ∠2 + ∠3 equals 90°. In triangle WXY, by the triangle angle sum theorem, we know ∠2 + ∠WYX + ∠WXY = 180°.

2. Substituting ∠WXY = 90°, ∠WYX = 38°, we get ∠2 = 52°.

3. Since ∠2 + ∠3 equals 90°, then ∠3 = 90° - 52° = 38°.

4. Finally, through the above reasoning, we conclude ∠3 = 38°.

矩形; 内错角; 直角三角形; 三角形的外角; 垂线

Triangle Angle Sum Theorem

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

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


Question: Place a ruler and a triangle board as shown in the figure, ∠1=40° then the degree of ∠2 is ()

Answer: 130°

1. Given ∠BAC=90°, and ∠BCA and ∠1 are the two angles that form a right angle. According to the property of complementary acute angles in a right triangle, i.e., ∠1+∠BCA=90°, we get ∠BCA=90°-∠1.

2. Based on the given condition, ∠1=40°, we get ∠BCA=90°-40°=50°.

3. According to the definition of adjacent supplementary angles, we get ∠BCE=180°-∠BCA=130°.

4. Because the two edges of the ruler are parallel, according to the property of corresponding angles being equal in parallel lines, we get ∠2=∠BCE=130°.

5. Through the above reasoning, the final answer is 130°.

Definition of Parallel Lines

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

<note>The long straight lines on the ruler are located in the same plane</note>, and they have no intersection points, so according to the definition of parallel lines, <note>the long straight lines on the ruler</note> are parallel lines.

Parallel Postulate 2 of Parallel Lines

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

<note>Two parallel lines (ruler)</note> intersected by <note>the long right-angle side BC of the triangle ruler</note> form the following geometric relationship: 1. Corresponding angles: <note>∠2 and ∠BCE are equal</note>.

Triangle Angle Sum Theorem

<note>In the figure of this problem, in triangle BAC</note>, <note>angle BAC, angle BCA, and angle 1 are the three interior angles of triangle ABC</note>, according to the Triangle Angle Sum Theorem, <note>angle BAC + angle ACB + angle 1 = 180°</note>.

Definition of Corresponding Angles

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

<note>Two parallel lines x are intersected by a transversal</note>, where <note>∠2 and ∠BCE are on the same side of the transversal and on the same side of the intersected lines</note>, therefore <note>∠2 and ∠BCE are corresponding angles</note>. Corresponding angles are equal, i.e., <note>∠2=∠BCE</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>∠BCE and ∠BCA share a common side BC</note>, <note>their other sides are extensions in opposite directions</note>, so <note>∠BCE and ∠BCA are adjacent supplementary angles</note>.


Question: After some oil is filled into a cylindrical oil tank with a diameter of 200cm, the cross-section is shown in the figure. If the width of the oil surface AB=160cm, then the maximum depth of the oil is ()

Answer: 40cm

1. Connect ##OA, OB##, and draw a perpendicular line OE from the center O to the midpoint M of AB.

2. According to the perpendicular bisector theorem, the diameter is perpendicular to any chord and bisects the chord, so OM is perpendicular to AB and M is the midpoint of AB.

3. According to the problem, the length of AB is 160cm, and point M is the midpoint of AB, so AM = 80cm.

4. The radius of the circle is half of 200cm, that is, OA = 100cm.

5. In the right triangle OAM, according to the Pythagorean theorem, ##OM? = OA? - AM? = 100? - 80? = 3600?. OM = √3600 = 60##

####

##6##. Since the highest point of the oil surface is ##M##, the depth of the oil tank is the length of the radius ##OE minus OM##, so the maximum depth is 100cm - 60cm = 40cm.

8. Through the above reasoning, the final answer is that the maximum depth of the oil is 40cm.

圆; 弦; 直角三角形; 正弦; 垂线

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.

In the figure of this problem, <note>the diameter is AB</note>, <note>connecting the center O and points A and B on the circumference</note>, with a length that is twice the radius, that is, <note>AB = 2 * 100cm = 200cm</note>.

Definition of Radius

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

In the figure of this problem, <note>in circle O, point O is the center of the circle, points A and B are any points on the circle, line segments OA and OB are the segments from the center to any point on the circle</note>, therefore <note>line segments OA and OB are the radii of the circle</note>, and their length is 100 cm.

Definition of Chord

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

Original text: <note>In circle O, points A and B are any two points on the circle</note>, <note>line segment AB connects these two points</note>, so <note>line segment AB is a chord of circle O</note>.

Definition of Right Triangle

<note>In the diagram of this problem, in triangle OAM, angle ##∠OMA## is a right angle (90 degrees)</note>, therefore <note>triangle OAM is a right triangle</note>. <note>Side OA and side OM are the legs</note>, <note>side AM is the hypotenuse</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>Line OM and line AB intersect to form an angle ∠OMA of 90 degrees</note>, therefore according to the definition of perpendicular lines, <note>line OM and line AB are perpendicular to each other</note>.

Pythagorean Theorem

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

In the figure of this problem, in the right triangle OAM, <note>∠OMA is a right angle (90 degrees)</note>, <note>sides OM and AM are the legs</note>, <note>side OA is the hypotenuse</note>, so according to the Pythagorean Theorem, ##OM² = OA² - AM²##.

Perpendicular Diameter Theorem

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

<note>In circle O, diameter ##OE##</note> is perpendicular to <note>chord AB</note>, then according to the Perpendicular Diameter Theorem, <note>diameter ##OE## bisects chord AB</note>, that is, <note>AM=MB</note>, and <note>diameter ##OM## bisects the two arcs subtended by chord AB</note>, that is, <note>arc EA=arc EB</note>.


Question: Find the area of the figure (unit: feet). If necessary, round to the nearest tenth.

Answer: 34.6

1. Given that the quadrilateral in the figure has exactly one pair of parallel sides, according to the definition of a trapezoid, this quadrilateral is a trapezoid. The top base of the trapezoid is 12 inches and the bottom base is 8 inches.

2. From the two ends of the bottom base of the trapezoid, draw segments perpendicular to the base. These two segments are the height h of the trapezoid. According to the definition of a rectangle, the two heights and the bases they intersect form a rectangle, so the long side of the rectangle is 8 feet.

3. The height of the trapezoid and the two interior angles of the base form two right triangles. Given that the two slant sides of the trapezoid are equal, according to the congruence criteria of right triangles (hypotenuse and one leg), the two right triangles are congruent. Let the short leg of these two right triangles be x, then the base is 2x + 8 = 12, solving for x gives x = 2.

4. One angle of the two right triangles is 30°, so these two right triangles are both 30°-60°-90° triangles. According to the properties of 30°-60°-90° triangles, the long leg is √3 times the short leg, therefore h = 2√3 ≈ 3.464.

5. The area A of the trapezoid can be calculated using the formula, A = 0.5 × (top base + bottom base) × height, i.e., A = 0.5 × (12 + 8) × 3.464.

6. Using a calculator, A ≈ 34.64 square inches.

7. Rounding the result to the nearest tenth, the area of the figure is 34.6 square inches.

梯形; 直角三角形; 正弦

Definition of Trapezoid

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

In the figure of this problem, in the quadrilateral, <note>the top and bottom sides are parallel</note>, while <note>the other two sides are not parallel</note>. Therefore, according to the definition of a trapezoid, this quadrilateral is a trapezoid because <note>it has exactly one pair of parallel sides</note>.

Definition of Right Triangle

In the figure of this problem, the two perpendicular lines of the trapezoid form two <note>right triangles</note> with the base and the top base respectively. The <note>right angles of these two right triangles</note> are located at the intersection points of the perpendicular lines with the base and the top base, and the two <note>acute angles are 30 degrees and 60 degrees respectively</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>The 30-degree angle is the angle on the left side of the trapezoid</note>, <note>the 60-degree angle is the other acute angle of the right triangle</note>, <note>the 90-degree angle is the right angle of the right triangle</note>. <note>Side x is the side opposite the 30-degree angle</note>, <note>side h is the side opposite the 60-degree angle</note>. According to the properties of a 30°-60°-90° triangle, <note>side h is equal to side x multiplied by √3</note>. That is: <note>x = 2 inches</note>, <note>h = x√3 = 2√3</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 12 inches</note>, <note>the lower base is 8 inches</note>, <note>the height h is approximately 3.464 inches</note>. According to the trapezoid area formula, Area = 0.5 * (12 + 8) * 3.464.

Definition of Rectangle

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

<note>The two heights of the trapezoid and the two bases cut by the heights form a rectangle</note>, <note>all four interior angles are right angles (90 degrees)</note>, and <note>the two long sides are parallel and equal in length</note>, <note>the two heights h are parallel and equal in length</note>.

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

<note>The two angles formed by the altitude perpendicular to the base in the two triangles are both right angles (90 degrees)</note>. <note>The two hypotenuses are equal</note>, so according to the Right Triangle Congruence (Hypotenuse, One Leg), these two right triangles are congruent.

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).

In the figure of this problem, <note>the line segments drawn perpendicular to the base from both ends of the lower base of the trapezoid are the height of the trapezoid</note>. <note>The two line segments form a right angle (90 degrees) with the base</note>, which indicates that the two line segments are the vertical distance from the vertex to the opposite side.


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

Answer: 4

1. Given that quadrilateral PQRS is a parallelogram, according to the properties of parallelograms, the diagonals of a parallelogram bisect each other.

2. ##Therefore QT=ST, as shown in the figure, QT=5y and ST=2y+12, so 5y=2y+12##.

3. ##Solve the equation: 5y=2y+12, move 2y to the left side of the equation to get: 5y-2y=12, so 3y=12, finally solving: y=4##.

####

##4. Therefore, when the quadrilateral is a parallelogram, y=4. Conversely, when y=4, the quadrilateral will be a parallelogram##.

平行四边形; 平行线; 对顶角; 同旁内角

Definition of Parallelogram

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

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

Properties of Parallelogram Theorem

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

<note>In the figure of this problem, in the parallelogram PQRS</note>, <note>the opposite angles ∠QRS and ∠SPQ are equal</note>, <note>the opposite angles ∠PQR and ∠RSP are equal</note>; <note>sides PQ and RS are equal</note>, <note>sides QR and SP are equal</note>; <note>the diagonals PR and QS bisect each other</note>, that is, <note>the intersection point T divides the diagonal PR into two equal segments PT and TR</note>, <note>and divides the diagonal QS into two equal segments QT and TS</note>.


Question: As shown in the figure, an exterior angle ∠DCE of quadrilateral ABCD is 70°, then the measure of ∠BAD is ()

Answer: 70°

1. Given that quadrilateral ABCD is a cyclic quadrilateral of circle O, according to the ##(Cyclic Quadrilateral Theorem Corollary 3) Opposite Angles Supplementary Theorem##, the opposite angles of the quadrilateral are supplementary.

2. According to the given conditions, the exterior angle ∠DCE = 70° ##therefore, ∠BCD = 180° - ∠DCE = 180° - 70° = 110°##.

3. ##According to the Cyclic Quadrilateral Theorem Corollary 3, ∠DAB + ∠BCD = 180°##.

5. In conclusion, through the properties of the cyclic quadrilateral, we obtain ∠BAD = 70°.

圆内接四边形; 三角形的外角; 圆周角; 邻补角

Cyclic Quadrilateral

A quadrilateral is called a cyclic quadrilateral if and only if all four vertices lie on the circumference of a single circle. This circle is referred to as the circumcircle of the quadrilateral.

<note>Quadrilateral ABCD has four vertices A, B, C, and D all on the same circle O</note>. This circle is called <note>the circumcircle of quadrilateral ABCD</note>. Therefore, <note>quadrilateral ABCD is a cyclic quadrilateral</note>.

In a cyclic quadrilateral (a quadrilateral inscribed in a circle), the sum of each pair of opposite angles is equal to 180°. Additionally, any exterior angle is equal to its interior opposite angle.

<note>Quadrilateral ABCD is a cyclic quadrilateral</note>, according to the Diagonal Supplementary Theorem for Cyclic Quadrilateral, <note>the sum of each pair of opposite angles of quadrilateral ABCD is equal to 180°</note>. Specifically, <note>∠BAD + ∠BCD = 180°</note>; <note>∠ABC + ∠ADC = 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.

In the figure of this problem, <note>angle BCD and angle DCE share a common side CD</note>, and their other sides <note>CB and CE are extensions in opposite directions</note>, so <note>angle BCD and angle DCE are adjacent supplementary angles</note>.


Question: As shown in the figure, A, B, C are three points on ⊙O, ∠ACB=25°, then the degree of ∠BAO is ()

Answer: 65°

1. Connect OA and OB, because we need to use them to analyze the angle relationships.

2. Connect OB and OA, according to the problem statement we know ∠ACB = 25°.

3. According to the inscribed angle theorem, the measure of the inscribed angle is half of the central angle it subtends. Therefore, the central angle ∠AOB = 2 * ∠ACB = 2 * 25° = 50°.

4. Since OA and OB are radii of the circle, their lengths are equal, so OA = OB. Therefore, △OAB is an isosceles triangle.

5. In an isosceles triangle, the two base angles are equal, so we set ∠OAB = ∠OBA.

6. In △OAB, since it is a triangle, the sum of the interior angles is 180°, so ∠OAB + ∠OBA + ∠AOB = 180°.

7. Substitute the known conditions and angle values, we have ∠OAB + ∠OAB + 50° = 180°.

8. Simplify to get 2∠OAB = 180° - 50°, that is 2∠OAB = 130°.

9. Continue to simplify to get ∠OAB = 65°.

圆; 圆周角; 圆心角; 弧; 点

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>vertex C of angle ACB is on the circumference</note>, <note>the two sides of angle ACB intersect circle O at points A and B respectively</note>. Therefore, <note>angle ACB 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 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 Isosceles Triangle

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

In the triangle OAB, sides OA and OB are equal, therefore the triangle OAB is an isosceles triangle.

Triangle Angle Sum Theorem

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

<note>Angle OAB, angle OBA, and angle AOB are the three interior angles of triangle OAB</note>. According to the Triangle Angle Sum Theorem, <note>angle OAB + angle OBA + angle AOB = 180°</note>.

Inscribed Angle Theorem

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

<note>In circle O, points A, B, and C are on the circle</note>, the central angle corresponding to arc AC and arc CB is <note>∠AOB</note>, and the inscribed angle is <note>∠ACB</note>. According to the Inscribed Angle Theorem, <note>∠ACB is equal to half of the central angle ∠AOB corresponding to arc AC</note>, i.e., <note>∠ACB = 1/2 ∠AOB</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 OAB, sides OA and OB are equal</note>. Therefore, according to the properties of isosceles triangles, the angles opposite the equal sides are equal, i.e., <note>angle OAB = angle OBA</note>.


Question: Find x.

Answer: 16

1. According to the given conditions, in triangle LMN, ∠MLN = 65°, ∠LMN = 51°, we can use the triangle angle sum theorem to find ∠LNM.

2. The triangle angle sum theorem states that the sum of the interior angles of a triangle is 180°. Therefore, ∠LNM = 180° - ∠MLN - ∠LMN.

3. Calculation gives: ∠LNM = 180° - 65° - 51° = 64°.

4. ##Given ∠MLN=∠YZX, ∠LMN=∠ZYX, according to the similarity criterion of triangles (AA), we can conclude that triangle XYZ is similar to triangle LMN.##

5. ##According to the definition of similar triangles, we can conclude that ∠LNM=∠ZXY=(4x)°=64°.##

6. Therefore, we can get 4x = 64. Solving this equation will give the value of x.

8. Simplifying the calculation, we get x = 16.

普通三角形; 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.

In the figure of this problem, <note>triangle LMN</note> is a geometric figure composed of <note>three non-collinear points L, M, N</note> and their <note>connecting line segments LM, MN, NL</note>. <note>Points L, M, N</note> are the <note>three vertices</note> of the triangle, and <note>line segments LM, MN, NL</note> are the <note>three sides</note> of the triangle. <note>Triangle XYZ</note> is a geometric figure composed of <note>three non-collinear points X, Y, Z</note> and their <note>connecting line segments XY, YZ, ZX</note>. <note>Points X, Y, Z</note> are the <note>three vertices</note> of the triangle, and <note>line segments XY, YZ, ZX</note> are the <note>three sides</note> of the triangle.

Similarity Theorem for Triangles (AA)

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

<note>Angle LMN is equal to angle ZYX</note>, and <note>angle MLN is equal to angle YZX</note>, so <note>triangle LMN is similar to triangle XYZ</note>.

Triangle Angle Sum Theorem

In <note>triangle LMN</note>, angle MLN, angle LMN, and angle LNM are <note>the three interior angles of triangle LMN</note>. According to the Triangle Angle Sum Theorem, <note>angle MLN + angle LMN + angle LNM = 180°</note>.

Definition of Similar Triangles

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

<note>Triangles XYZ and LMN are similar triangles</note>. According to the <note>definition of similar triangles</note>, we have: <note>∠MLN = ∠YZX</note>, <note>∠LMN = ∠ZYX</note>, <note>∠LNM = ∠ZXY</note>; <note>ML/YZ = LN/ZX = MN/YX</note>.


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

Answer: 528

1. Let parallelogram ABCD, as shown in the figure, AE⊥BC. According to the definition of the height of a parallelogram, AE is the height of parallelogram ABCD. According to the area formula of a parallelogram A = base length * height.

2. Since the base BC length is 22 cm and AE = 24 cm, substitute into the formula to get A = 22 cm * 24 cm.

3. Perform the calculation to get A = 528 square centimeters.

4. Finally, the area of the parallelogram is 528 square centimeters, no rounding needed.

平行四边形; 平行线; 垂线

Definition of Altitude

<note>In parallelogram ABCD, side AD and side BC are parallel</note>. <note>The perpendicular distance from vertex A to the opposite side BC is the altitude of the parallelogram</note>. Specifically, <note>draw a line segment AE from vertex A perpendicular to side BC</note>, <note>the length of this line segment AE is the altitude of the parallelogram</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 \( A = \text{base length} \times \text{height} \).

<note>The base BC of parallelogram ABCD</note> is 22 cm long, <note>the corresponding height AE is the vertical distance from the base BC to the opposite side</note>, which is 24 cm. Therefore, according to the area formula of a parallelogram, the area of the parallelogram is equal to <note>the base length 22 cm</note> multiplied by <note>the corresponding height 24 cm</note>, i.e., A = 22 cm × 24 cm = 528 square centimeters.


Question: As shown in the figure, AB is the diameter of ⊙O, point C is a point on ⊙O, ∠C=20°, then the degree of ∠BOC is ()

Answer: 40°

1. Given AB is the diameter of ⊙O, point C is on ⊙O, ∠ACB=20°.

2. Since OA=OC, △OAC is an isosceles triangle, thus ∠OCA=∠OAC=20°.

3. According to the inscribed angle theorem, the inscribed angle is half of its corresponding central angle, here ##the inscribed angle BAC is half of the central angle BOC##. Therefore, ##∠BOC=2∠BAC##.

4. Substituting the given angle, ##∠BOC##=2×20°=40°.

5. The final correct answer is 40°.

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 2 times the radius, that is <note>AB = 2×OA</note>.

Definition of Central Angle

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

Definition of Inscribed Angle

The vertex of <note>angle BAC</note>, <note>A</note>, is on the circumference of the circle, and the two sides of <note>angle BAC</note> intersect with <note>circle O</note> at <note>point C</note> and <note>point B</note> respectively. Therefore, <note>angle BAC</note> is an inscribed angle.

Definition of Isosceles Triangle

<note>Side OA and side OC are equal</note>, therefore <note>triangle OAC is an isosceles triangle</note>.

Inscribed Angle Theorem

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

Properties of Isosceles Triangle

In the isosceles triangle <note>OAC</note>, the sides <note>OA and OC are equal</note>. Therefore, according to the properties of isosceles triangles, the angles opposite the equal sides are equal, that is, <note>∠OAC = ∠OCA</note>.


Question: As shown in the figure, AB∥CD, AD bisects ∠BAC, and ∠C=80°, then the degree of ∠D is ()

Answer: 50°

1. Given AD bisects ∠BAC, ∴ ∠BAD = ∠CAD.

2. According to the parallel axiom 2 of parallel lines and the definition of alternate interior angles, AB∥CD, parallel lines AB and CD are intersected by line AD, so alternate interior angles are equal, and ∠D and ∠BAD are alternate interior angles, so ∠D=∠BAD. Also, since AD bisects ∠BAC, ∠D=∠BAD = ∠CAD.

3. Based on the above conclusion, we get ∠CAD = ∠D.

4. In △ACD, according to the triangle angle sum theorem, ∠C + ∠CAD + ∠D = 180°.

5. Substitute the given ∠C = 80° and ∠CAD = ∠ADC, i.e., 80° + ∠D + ∠D = 180°.

6. Solve the above equation, we get ∠D = 50°.

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

平行线; 内错角; 普通三角形

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 BAC is point A</note>, a line AD is drawn from <note>point A</note>, <note>this line divides angle BAC into two equal angles</note>, that is, <note>∠BAD and ∠CAD are equal</note>. Therefore, <note>line AD is the angle bisector of angle BAC</note>.

Definition of Parallel Lines

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

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

Parallel Postulate 2 of Parallel Lines

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

In the diagram of this problem, <note>two parallel lines AB and CD</note> are intersected by <note>the third line AD</note>, forming the following geometric relationships:
1. Corresponding angles: ##none##.
2. Alternate interior angles: <note>∠BAD and ∠D are equal</note>.
3. Same-side interior angles: ##none##.
These relationships illustrate that <note>when two parallel lines are intersected by a third line, the alternate interior angles are equal</note>.

Triangle Angle Sum Theorem

<note>Angles ∠CAD, ∠ADC, and ∠ACD are the three interior angles of triangle ACD</note>. According to the Triangle Angle Sum Theorem, <note>∠CAD + ∠ADC + ∠ACD = 180°</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.

In the diagram of this problem, two parallel lines <note>AB</note> and <note>CD</note> are intersected by a line <note>OD</note>, where <note>angle D</note> and <note>angle AOD</note> are located between the two parallel lines and on opposite sides of the intersecting line <note>OD</note>. Therefore, <note>angle D and angle AOD are alternate interior angles</note>. Alternate interior angles are equal, i.e., <note>angle D is equal to angle AOD</note>.


Question: As shown in the figure, the lines l∥m∥n, the vertices B and C of triangle ABC are on lines n and m respectively, the angle between side BC and line n is 25°, and ∠ACB=60°, then the degree of ∠a is ()

Answer: 35°

1. Given that the lines l∥m∥n, and point B is on line n, point C is on line m, therefore the angle between BC and line n is 25°.

2. Let the angle between AC and line m be ∠b, and the angle between BC and line m be ∠c. According to the parallel line axiom 2 and the definition of corresponding angles, corresponding angles between parallel lines are equal, so ∠a is equal to ∠b, i.e., ∠a = ∠b.

3. According to the parallel line axiom 2 and the definition of alternate interior angles, alternate interior angles are equal, and the angle between BC and line n and ∠c are alternate interior angles, so ∠c=25°. According to the problem statement, ∠ACB=60°, ∠a = ∠b, and ∠ACB = ∠b + ∠c, so ∠ACB = ∠a + ∠c.

4. From the previous step, we get ∠ACB = 25° + ∠a = 60°, thus ∠a = 60° - 25°.

5. Calculating, we get ∠a = 35°.

6. Through the above reasoning, we finally obtain the degree of ∠a as 35°.

平行线; 三角形的外角; 内错角; 同位角; 普通三角形

Definition of Parallel Lines

<note>Line l and line m lie in the same plane and they do not intersect</note>, therefore according to the definition of parallel lines, <note>line l and line m are parallel lines</note>. <note>Line n and line m lie in the same plane and they do not intersect</note>, therefore according to the definition of parallel lines, <note>line n and line m are parallel lines</note>.

Parallel Postulate 2 of Parallel Lines

In the figure of this problem, ##two parallel lines <note>l and m</note> are intersected by a third line <note>AC</note>, forming the following geometric relationships: 1. Corresponding angles: <note>angle a and angle b are equal</note>. 2. Alternate interior angles: none. 3. Same-side interior angles: none. These relationships indicate that when two parallel lines are intersected by a third line, the corresponding angles are equal. Two parallel lines <note>m and n</note> are intersected by a third line <note>BC</note>, forming the following geometric relationships: 1. Corresponding angles: none. 2. Alternate interior angles: <note>angle c and the 25° angle are equal</note>. 3. Same-side interior angles: none. These relationships indicate that when two parallel lines are intersected by a third line, the alternate interior angles 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 l and m are intersected by line AC</note>, where <note>angle ∠a and angle ∠b are on the same side of the intersecting line AC</note>, <note>on the same side of the two intersected lines l and m</note>, therefore <note>angle ∠a and angle ∠b are corresponding angles</note>. <note>Corresponding angles are equal, that is, angle ∠a is equal to angle ∠b</note>.

Definition of Alternate Interior Angles

<note>Two parallel lines m and n</note> are intersected by <note>a transversal BC</note>, where <note>angle c and the 25° angle are located between the two parallel lines and on opposite sides of the transversal BC</note>. Therefore, <note>angle c and the 25° angle are alternate interior angles</note>. <note>Alternate interior angles are equal</note>, that is, <note>angle c is equal to the 25° angle</note>.


Question: As shown in the figure, Xiao Li, a student with a height of 1.6 meters, wants to measure the height of the flagpole at school. When he stands at point C, the top of his shadow coincides with the top of the flagpole's shadow, and he measures AC=2 meters and BC=8 meters. What is the height of the flagpole?

Answer: 8米

1. Let the height of the flagpole be h meters.

2. According to the problem, the shadow of Xiao Li's head coincides with the shadow of the flagpole's top. Let the top of the flagpole be E, and the top of Xiao Li's head intersect AE at point D. According to the theorem of similar triangles (AA), ∠ACD=ABE=90°, ∠DAC=EAB, so △ADC∽△ABE.

3. According to the definition of similar triangles, we have AC/AB=DC/BE=AD/AE.

4. Given AC=2, CB=8, AB=AC+BC=10, CD=1.6.

5. Substitute the known values into the ratio, we get (2 meters)/(8 meters + 2 meters) = 1.6 meters/BE.

6. Simplify the ratio, we get: 1.6 meters/BE = 2 meters/10 meters.

7. Solve the ratio, we get: BE = 8 meters.

8. Through the above reasoning, we finally conclude that the height of the flagpole is 8 meters.

Definition of Similar Triangles

<note>Triangle ACB and triangle DCE are similar triangles</note>. According to the definition of similar triangles, we have: <note>∠ACD = ∠ABE</note>, <note>∠ADC = ∠AEB</note>, <note>∠DAC = ∠EAB</note>; <note>AC/AB = DC/BE</note>, that is <note>2/10 = 1.6/BE</note>, we can derive the proportion <note>2/10 = 1.6/BE</note>, and finally find <note>BE = 8 meters</note>.

Similarity Theorem for Triangles (AA)

In the given diagram, in triangles ADC and ABE, if <note>angle ACD is equal to angle ABE</note> and <note>angle DAC is equal to angle EAB</note>, then <note>triangle ADC is similar to triangle ABE</note>.


Question: Find AB.

Answer: 23

1. Given conditions: side AB = 2x + 5, side BC = 3x - 4, side AC = 27. There are red marks on the segments AB and BC indicating that they are equal, ##i.e., AB = BC##.

####

##2##. Therefore, we can deduce 2x + 5 = 3x - 4.

##3##. Solve this equation: move 3x to the left side of the equation and move the constants to the right side, resulting in 2x - 3x = -4 - 5.

##4##. Simplify the equation to get -x = -9, hence x = 9.

##5. Substitute x = 9 into the expression for side AB, obtaining AB = 2x + 5 = 2*9 + 5 = 18 + 5 = 23.##

##6##. Through the above reasoning, the final answer is AB = 23.

Definition of Isosceles Triangle

In <note>triangle ABC</note>, <note>sides AB and BC are equal</note>, therefore <note>triangle ABC is an isosceles triangle</note>.

Properties of Isosceles Triangle

<note>In the diagram of this problem, in the isosceles triangle ABC, side AB and side BC are equal</note>. Therefore, according to the properties of isosceles triangles, the angles opposite the equal sides are equal, that is, <note>∠BAC = ∠BCA</note>.


Question: As shown in the figure, a sector-shaped paperboard with a radius of 24cm is used to make a conical hat (seam is ignored). If the radius of the base of the conical hat is 10cm, then the area of this sector-shaped paperboard is ()

Answer: 240πcm^{2}

1. Given that the base radius of the conical hat is 10cm, and the radius of the sector paper used is 24cm, ##let a point on the base circle be point A, the vertex of the cone be point B, and connect AO, then AO is the radius of the circle##.

2. ##The radius of the sector is the slant height of the cone##, that is, the slant height of the conical hat is 24cm.

3. According to the surface area formula of the cone, the lateral area is equal to the area of the sector, and the lateral area is equal to πrl, that is, π × base circle radius × slant height. Therefore, the area formula of the sector is: S = π × R × l = π × 10 × 24 = 240 π cm^2.

4. Through the above reasoning, the final answer is 240π cm^2.

扇形; 圆锥; 圆; 弧; 弦

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>The base of the conical hat is a circle, and the center of the circle is O</note>, <note>any point on the circle is A</note>, <note>the line segment OA is the line segment from the center of the circle to any point on the circle, so the line segment OA is the radius of the circle</note>, and OA = 10cm.

Generatrix

The generatrix of a cone is the line segment that joins a point on the circumference of the base to the apex.

<note>In the figure of this problem, in the cone, a point A on the circumference of the base and the vertex B are connected by the line segment AB, which is the generatrix</note>. The generatrix is the line segment from a point on the circumference of the base to the vertex in a cone.

Formula for the Surface Area of a Cone

The total surface area of a cone is equal to the sum of the base area and the lateral surface area.

In this problem diagram, in the cone, the base is a circle with a <note>radius of 10</note> and a <note>base area of π×10²</note>. The lateral surface of the cone, when unfolded, is a sector with a <note>radius of slant height 24</note> and a <note>sector arc length equal to the circumference of the base 2π×10</note>. The lateral area is equal to the area of the sector, which is <note>π×10×24</note>. The total surface area of the cone is equal to the base area plus the lateral area, so the <note>total surface area is π×10² + π×10×24</note>.


Question: This problem examines the inscribed angle theorem. The key is to solve it based on the relationship between the central angle and the inscribed angle subtended by the same chord. 4. As shown in the figure, AB is the diameter of ⊙O, and C is a point on ⊙O (excluding A and B). If ∠AOD = 136°, then the measure of ∠C is ()

Answer: 22°

1. Given AB is the diameter of ⊙O, ∠AOD=136°.

2. According to the definition of a straight angle, on the line AB, ∠AOD + ∠BOD = 180°.

3. Thus, we can find ∠BOD = 180° - 136° = 44°.

4. According to the Inscribed Angle Theorem, the inscribed angle ∠BCD subtended by arc BD is equal to half of the central angle ∠BOD subtended by arc BD, i.e., ∠BCD = ∠BOD / 2.

5. Specifically in this problem, the central angle ∠BOD subtended by arc BD = 44°.

6. Therefore, the inscribed angle ∠BCD subtended by arc BD = 44° / 2 = 22°.

####

##7##. After the above reasoning, the final answer is that the measure of ∠C is 22°.

圆; 圆心角; 圆周角; 弦; 直角三角形

Definition of Circle

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

<note>The center of the circle in the diagram is O</note>. In circle O, <note>point O is the center</note>. All points in the diagram that are at a distance equal to the radius from <note>point O are on circle O</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.

In the figure of this problem, in the circle <note>O</note>, point <note>B</note> and point <note>D</note> are two points on the circle, and the center of the circle is point <note>O</note>. The angle <note>∠BOD</note> formed by the lines <note>OB</note> and <note>OD</note> is called the central angle.

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, <note>the vertex of angle BCD (point C) is on the circumference of circle O</note>, <note>the two sides of angle BCD intersect circle O at points B and D respectively</note>. Therefore, angle BCD is an inscribed angle.

Inscribed Angle Theorem

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

<note>The central angle corresponding to arc BD is ∠BOD</note>. According to the Inscribed Angle Theorem, <note>the inscribed angle ∠BCD corresponding to arc BD is half of ∠BOD, i.e., ∠BCD = 1/2 ∠BOD</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.

<note>In the figure of this problem, ray OA rotates around endpoint O to form a straight line with the initial side, creating a straight angle AOB</note> According to the definition of a straight angle, <note>a straight angle measures 180 degrees, i.e., angle AOB = 180 degrees</note>.