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

Answer: 0.92

1. In the right triangle PMN, ##∠NPM## is a right angle. According to the properties of right triangles, ##sin(∠N)## represents the ratio of the opposite side PM to the hypotenuse MN.

2. The length of the opposite side PM of the right triangle PMN is 36, and the length of the hypotenuse MN is 39, which can be directly read from the given data.

3. From the sin Definition (i.e., ##sine function definition##), we get: ##sin(∠N)## = PM / MN.

4. Therefore, ##sin(∠N)## = 36 / 39.

5. Simplify 36 / 39 and express it in decimal form: 36 ÷ 39 ≈ 0.923.

6. According to the above calculation, the fourth step should adhere to the precision ##to two decimal places##, resulting in 0.92.

##7##. Through the above reasoning, the final answer is ##sin∠N≈ 0.923##.

Definition of Sine Function

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

In the right triangle PMN, ##∠N## is an acute angle, side PM is the opposite side of angle ##∠N##, side MN is the hypotenuse. According to the definition of the sine function, the sine value of ##∠N## equals the ratio of the opposite side PM to the hypotenuse MN, that is sin(##∠N##) = PM / MN = 36 / 39.

Definition of Right Triangle

<note>∠NPM</note> is a right angle (90 degrees), so triangle PMN is a right triangle. Side <note>PN</note> and side <note>PM</note> are the legs, and side <note>MN</note> is the hypotenuse.


Question: As shown in the figure, in quadrilateral ABCD, E and F are the midpoints of AB and AD respectively. If EF=2, BC=5, CD=3, then tanC equals ()

Answer: \frac{4}{3}

1. In quadrilateral ABCD, E and F are the midpoints of sides AB and AD respectively, and BD is drawn as an auxiliary line.

2. Since E and F are the midpoints of AB and AD respectively, in triangles AEF and ABD, AE:AB=1:2, AF:AD=1:2, and the two triangles share ∠A. According to the similarity theorem (SAS), triangles AEF and ABD are similar.

3. According to the definition of similar figures, EF:BD=1:2. Given EF=2, then BD=EF*2=4.

4. Since BD=4, CD=3, BC=5, according to the converse of the Pythagorean theorem, BD, CD, and BC satisfy BD^2 + CD^2 = BC^2, so triangle BCD is a right triangle with the right angle at point D.

5. In right triangle BCD, tan∠BCD equals the ratio of BD to CD.

6. Therefore, tan∠BCD = BD / CD = 4 / 3.

7. Through the above reasoning, it is concluded that tan∠C equals 4/3.

中点; 普通四边形; 线段; 正切

Midpoint of a Line Segment

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

<note>The midpoint of line segment AB is point E</note>. According to the definition of the midpoint of a line segment, point E divides line segment AB into <note>two equal parts</note>, that is, <note>the lengths of line segments AE and EB are equal</note>. That is, <note>AE = EB</note>. Similarly, <note>the midpoint of line segment AD is point F</note>, point F divides line segment AD into <note>two equal parts</note>, that is, <note>the lengths of line segments AF and FD are equal</note>. That is, <note>AF = FD</note>.

Definition of Tangent Function

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

<note>In right triangle BCD</note>, <note>angle ∠BCD is an acute angle</note>, <note>side BD is the opposite side of angle ∠BCD</note>, <note>side CD is the adjacent side of angle ∠BCD</note>, so <note>the tangent value of angle ∠BCD is equal to the length of side BD divided by the length of side CD</note>, that is, <note>tan(∠BCD) = BD / CD = 4 / 3</note>.

SAS Criterion for Similar Triangles

If two triangles have two pairs of corresponding sides in proportion and the included angle between those sides is equal, then the two triangles are similar.

<note>In triangles AEF and ABD</note>, <note>side AE corresponds to side AB</note>, <note>side AF corresponds to side AD</note>, and <note>side AE/side AB = side AF/side AD</note>, and <note>the two triangles share ∠A</note>, so according to the <note>SAS Criterion for Similar Triangles</note>, <note>triangle AEF is similar to triangle ABD</note>.

Converse of the Pythagorean Theorem

If the sum of the squares of two sides of a triangle is equal to the square of the third side, then the triangle is a right triangle, and the angle opposite to the longest side is a right angle.

<note>The three sides of triangle BCD are BD, CD, and BC</note>, and satisfy <note>BD^2 + CD^2 = BC^2</note>. According to the converse of the Pythagorean Theorem, <note>triangle BCD is a right triangle</note>, <note>the angle opposite the longest side BC, angle BDC, is a right angle</note>.

Definition of Similar Figures

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

<note>In the figure of this problem, triangle AEF and triangle ABD are similar</note>. According to the definition of similar figures, their <note>corresponding sides are proportional</note>, that is <note>EF:BD=AF:AD=AE:AB</note>.


Question: As shown in the figure, a circular piece of paper and a sector-shaped piece of paper are cut out, so that they can just form a cone model. If the radius of the circle is 1 and the central angle of the sector is 90°, then the radius of the sector is ()

Answer: 4

1. Given that the radius of the circle is 1, according to the definition, the circumference of the circle can be expressed as 2π.

2. Let the radius of the sector be r, and the central angle of the sector be equal to 90°. ##Let the rectangle be ABCD (starting from the top left and taking vertices counterclockwise), from top to bottom, the arc lengths of the sector intersect the rectangle at X and Y##

3. According to the formula for the arc length of a sector, ##the arc length L of the sector is equal to the central angle θ (expressed in radians) multiplied by the radius r: L = θr, therefore here θ = 90°, using the conversion formula between degrees and radians, radians = 90*(π/180) = π/2##.

4. Substituting the given conditions, the arc length of the sector can be expressed as L = ##πr/2##.

####

##5##. Since the problem description states that the arc length of the sector is equal to the circumference of the base circle, we have (1/2)πr = 2π.

##6##. Solving this equation gives: r = 2π / ((1/2)π) = 2π * (2/π) = 4.

##7##. Through the above reasoning, we finally conclude that the radius of the sector is 4.

圆; 扇形; 圆心角; 圆锥

Circumference Formula of Circle

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

In Figure 1 of this problem, <note>the radius of the circle is 1</note>. According to the circumference formula of the circle, <note>the circumference C of the circle is equal to 2π times the radius r</note>, that is, C=2πr. Therefore, <note>C=2π×1=2π</note>.

Definition of Sector

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

<note>In this problem, the central angle of the sector is equal to 90°</note>, the two radii are respectively <note>the radius of the sector r and the distance from the center to the other endpoint of the arc length</note>, <note>the arc is the arc length of the sector</note>. Therefore, according to the definition of the sector, <note>the figure formed by these two radii and the arc they enclose is a sector</note>.

Formula for the Length of an Arc of a Sector

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

<note>The radius of the sector is r</note>, <note>the central angle θ = 90°</note>. According to the formula for the length of an arc of a sector, the arc length L equals the central angle θ multiplied by the radius r, where <note>θ is expressed in radians</note>, that is, <note>θ = 90° * (π/180) = π/2</note>. Therefore, <note>L = (π/2) * r</note>.

Development of a Cone

The development (or net) of a cone is a sector of a circle, where the radius of the sector is the slant height of the cone, and the arc length of the sector is equal to the circumference of the cone's base.

<note>The circular piece of paper cut out will serve as the base of the cone</note>, <note>The circumference of the base circle 2π</note> must be equal to <note>The arc length of the sector (1/2)πr</note>. Therefore, <note>(1/2)πr = 2π</note>.

Formula for Conversion between Degrees and Radians

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

<note>The central angle of the sector is 90°</note>, according to the formula for conversion between degrees and radians, the conversion to radians = degrees * (π/180), which is <note>90 * (π/180)</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 diagram of this problem, <note>in circle B</note>, <note>points X and Y are two points on the circle</note>, <note>the center of the circle is point B</note>. <note>The angle ∠XBY formed by the lines XB and YB is called the central angle</note>.


Question: Line AB is tangent to circle O. If AB = 8 and OB = 10, find the diameter of circle O.

Answer: 12

1. Given that AB is a tangent to circle O, ##OA## is the radius of circle O. According to the ##property of the tangent to a circle##, angle OAB is 90 degrees.

2. ##According to the definition of a right triangle, since ∠OAB=90°, triangle OAB is a right triangle. In right triangle OAB, according to the Pythagorean theorem, OA^2 + AB^2 = OB^2##.

3. ##Since AB=8 and OB=10, OA^2 + 8^2 = 10^2##.

4. Solving this equation, OA^2 + 64 = 100.

5. Moving 64 to the right side, we get OA^2 = 100 - 64.

6. Calculating, we get OA^2 = 36.

7. Taking the square root, we get OA = 6, which means the radius of the circle is 6.

8. Since the diameter is twice the radius, the diameter of the circle is 6 * 2 = 12.

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

直角三角形; 切线; 圆

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

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>A diameter is a line segment that passes through the center O of a circle and has its endpoints on the circumference</note>. For example, if <note>radius OA = 6</note>, then the diameter is <note>2 times OA</note>, which is <note>2 × 6 = 12</note>.

Definition of Right Triangle

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

In the figure of this problem, in triangle OAB, <note>angle OAB is a right angle (90 degrees)</note>, therefore <note>triangle OAB is a right triangle</note>. The sides <note>OA and AB are the legs</note>, and the side <note>OB is the hypotenuse</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>Circle O and line AB have only one common point A</note>, this common point is called the point of tangency. Therefore, <note>line AB is the tangent to circle O</note>.

Property of the Tangent Line to a Circle

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

In <note>circle O</note>, point <note>A is the tangent point of line AB to the circle</note>, and segment <note>OA is the radius of the circle</note>. According to the property of the tangent line to a circle, <note>the tangent line AB is perpendicular to the radius OA at the tangent point A</note>, i.e., <note>∠OAB=90 degrees</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 OAB, angle OAB is a right angle (90 degrees)</note>, <note>sides OA and AB are the legs, and side OB is the hypotenuse</note>, so according to the Pythagorean Theorem, ##<note>OB^2 = OA^2 + AB^2</note>##, that is 10^2 = OA^2 + 8^2. Through calculation, we can find OA^2 = 36, therefore OA = √36 = 6, that is <note>the radius of the circle is 6</note>. Finally, <note>the diameter is twice the radius</note>, so the diameter is 2×6=12.


Question: As shown in the figure, it is known that the inscribed angle ∠BAC=40°, then the central angle ∠BOC is () degrees.

Answer: 80°

1. Given the inscribed angle ∠BAC = 40°.

2. The arc corresponding to the central angle ∠BOC is the same as the arc corresponding to the inscribed angle ∠BAC.

3. According to the inscribed angle theorem, the relationship between the central angle ∠BOC and the inscribed angle ∠BAC is: the central angle is twice the inscribed angle, i.e., ∠BOC = 2∠BAC.

4. Substituting the given ∠BAC into the equation, we get ∠BOC = 2 × 40° = 80°.

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

圆; 圆周角; 圆心角; 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.

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

Inscribed Angle Theorem

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

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

Definition of Central Angle

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


Question: As shown in the figure, quadrilateral ABCD is inscribed in ⊙O, E is a point on the extension line of BC, ∠A=50°, then the degree of ∠DCE is ()

Answer: 50°

1. Given quadrilateral ABCD is inscribed in circle O, ∠A=50°, connect OB, OD.

2. ∠DOB and ∠A both subtend arc BD, according to the Inscribed Angle Theorem, we get ∠DOB=2∠A, ##i.e., ∠DOB=100°##.

3. ∠DOB and ∠BCD both subtend arc DAB, according to the Inscribed Angle Theorem, we get ∠DOB (angle greater than a straight angle)=2∠BCD.

4. ∠DOB (angle less than a straight angle) + ∠DOB (angle greater than a straight angle) = 360°, ##according to the definition of a complete angle, given ∠DOB=100°, so ∠DOB (angle greater than a straight angle) = 360° - ∠DOB (angle less than a straight angle) = 360° - 100° = 260°##.

5. ##Given ∠DOB (angle greater than a straight angle) = 2∠BCD, so ∠BCD = ∠DOB (angle greater than a straight angle) / 2 = 260° / 2 = 130°##.

6. From ∠DCE + ∠BCD = 180°, ##according to the definition of a straight angle, we get ∠DCE = 180° - ∠BCD##.

7. Because ##∠BCD = 130°##, so ∠DCE = 50°.

n. Through the above reasoning, the final answer is ∠DCE = 50°.

圆内接四边形; 圆周角; 邻补角; 对顶角

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 <note>circle ⊙O</note>, <note>point O is the center</note>, and the radius is <note>OA, OB, OC, OD</note>. All points in the figure that are at a distance equal to <note>OA, OB, OC, OD from point O</note> are on the <note>circle ⊙O</note>.

Definition of Inscribed Angle

In the figure of this problem, in circle O, <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>. Similarly, 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>.

Inscribed Angle Theorem

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

Definition of Full Angle

An angle formed when a ray revolves around its endpoint for one complete rotation is called a full angle.

In the problem diagram, ray OA rotates around endpoint O for a full circle, forming a full angle AOA. According to the definition of full angle, the degree of the full angle is 360 degrees, that is, full angle AOA = 360 degrees.

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>Ray BC rotates around endpoint C to form a straight line with the initial side</note>, forming a <note>straight angle BCE</note>. According to the <note>definition of straight angle, the measure of a straight angle is 180 degrees, i.e., angle BCE = 180 degrees</note>.

Definition of Central Angle

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


Question: Find CF, if BF bisects \angle ABC and AC \parallel ED, BA = 6, BC = 7.5, AC = 9, and DE = 9.

Answer: 5

1. Given ∠ABF=∠CBF, according to the definition of angle bisector, segment BF is the angle bisector of ∠ABC; according to the angle bisector theorem, we get AF/FC = AB/BC.

2. Substitute AB = 6, BC = 7.5, we get AF / FC = 6 / 7.5.

3. Convert 6/7.5 to a more precise decimal ratio, we get AF / FC = 4 / 5.

4. Set AF = 4k, FC = 5k, where k is the proportional factor.

5. Combining the total length AC = 9, we get 4k + 5k = 9, thus k = 1.

6. Solving for k=1, we get AF = 4 × 1 = 4, FC = 5 × 1 = 5.

7. Through the above reasoning, the final answer is CF = 5.

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 ABC is point B</note>, <note>a line BF is drawn from point B</note>, and this line divides angle ABC into two equal angles, that is, <note>angle ABF and angle CBF are equal</note>. Therefore, <note>line BF is the angle bisector of angle ABC</note>.

Angle Bisector Theorem

In a triangle, the angle bisector divides the opposite side into two segments that are proportional to the lengths of the other two sides of the triangle, such that the ratio between these segments is equal to the ratio of the other two sides.

In <note>triangle ABC</note>, the angle bisector of angle ABC, <note>BF</note>, intersects the opposite side <note>AC</note> at point <note>F</note>. The sides <note>AB</note> and <note>BC</note> are the two sides of angle ABC, and the segments <note>AF</note> and <note>FC</note> are the two parts into which the angle bisector <note>BF</note> divides the opposite side <note>AC</note>. According to the Angle Bisector Theorem, the ratio of side <note>AB</note> to side <note>BC</note> is equal to the ratio of segment <note>AF</note> to segment <note>FC</note>, that is, <note>AF / FC</note> = <note>AB / BC</note>.


Question: Find m \\angle 2.

Answer: 90

1. Observing the given figure, it can be found that the triangle in the figure is a right triangle, with the right angle located at the bottom right corner. The center of the circle is on the hypotenuse, which makes it part of the circle.

####

##2. According to (Corollary 2 of the Inscribed Angle Theorem) the inscribed angle subtended by the diameter is a right angle, the inscribed angle subtended by the semicircle (diameter) is a right angle; the chord subtended by a 90° inscribed angle is the diameter. Therefore, the given ∠2 is actually represented as 90°.

##3##. Through the above reasoning, the final answer is 90°.

圆; 圆周角; 直角三角形

Definition of Diameter

<note>The hypotenuse is the diameter</note>, connecting <note>the center of the circle and two points on the circumference</note>, with a length of <note>twice the radius</note>.

Definition of Inscribed Angle

<note>In the figure of this problem, in the circle, the vertex of angle 2 is on the circumference</note>, <note>the two sides of angle 2 intersect the circle at two points respectively</note>. Therefore, angle 2 is an inscribed angle.

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.

The original: <note>The hypotenuse of a right triangle is the diameter</note>, so <note>the inscribed angle 2 is 90 degrees</note>.


Question: Find Q M.

Answer: 12

1. Given the conditions, we need to find the length of segment QM, and there are two right angles in quadrilateral NPQM in the figure.

2. ##From the figure, we know ∠MNQ=∠PNQ, and since ∠NMQ and ∠NPQ are both right angles, the two triangles share the hypotenuse NQ. According to the congruence theorem (AAS), triangle MNQ≌triangle PNQ##.

3. ##According to the definition of congruent triangles, the corresponding sides of triangles MNQ and PNQ are equal, i.e., MQ=PQ##.

4. ##From the figure: MQ=2x+2, PQ=4x-8, substituting MQ=PQ gives the equation##: 4x - 8 = 2x + 2.

5. Solving the equation 4x - 8 = 2x + 2, we get 2x = 10, resulting in x = 5.

6. Substituting x = 5 into 2x + 2 to calculate the length of QM, the result is 2(5) + 2 = 10 + 2 = 12.

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

垂线; 直角三角形; 平行四边形

Definition of Right Triangle

In the figure of this problem, in triangle NMQ, <note>angle NMQ is a right angle (90 degrees)</note>, therefore <note>triangle NMQ is a right triangle</note>. <note>Side NM and side MQ are the legs</note>, <note>side QN is the hypotenuse</note>. Similarly, in triangle NPQ, <note>angle NPQ is a right angle (90 degrees)</note>, therefore <note>triangle NPQ is a right triangle</note>. <note>Side NP and side PQ are the legs</note>, <note>side QN is the hypotenuse</note>.

Congruence Theorem for Triangles (AAS)

Two triangles are congruent if two angles and the included side of one triangle are respectively equal to two angles and the included side of the other triangle.

In the diagram of this problem, in triangles <note>MNQ and PNQ</note>, <note>angle MNQ is equal to angle PNQ</note>, <note>angle NMQ is equal to angle NPQ</note>, and <note>side NQ is equal to NQ</note>. Since these two triangles have <note>two angles and the side opposite one of the angles equal</note>, according to the Congruence Theorem for Triangles <note>Angle-Angle-Side (AAS)</note>, it can be concluded that <note>triangle MNQ is congruent to triangle PNQ</note>.

Definition of Congruent Triangles

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

In the figure of this problem, <note>triangle MNQ and triangle PNQ are congruent triangles</note>, <note>the corresponding sides and corresponding angles of triangle MNQ and triangle PNQ are equal</note>, that is:
<note>side MN = side PN</note>
<note>side NQ = side NQ</note>
<note>side MQ = side PQ</note>
At the same time, the corresponding angles are also equal:
<note>angle MNQ is equal to angle PNQ</note>
<note>angle NMQ is equal to angle NPQ</note>
<note>angle MQN = angle PQN</note>.


Question: If A B \perp B C, find the degree measure of \angle 2.

Answer: 68

1. Let the triangle containing ∠2 be ADE, i.e., ∠ADE = 36°. 2. According to the figure in the problem, ∠AEC = 104°. 3. In triangle ADE, according to the exterior angle theorem: the exterior angle of any triangle is equal to the sum of the two non-adjacent interior angles. 4. Since ∠AEC is the exterior angle of ∠AED, and ∠AED = ∠1, therefore according to the exterior angle theorem, ∠AEC = ∠ADE + ∠DAE. Hence ∠2 = ∠DAE = ∠AEC - ∠ADE. 5. Substituting the known values of ∠AEC and ∠ADE into ∠2 = ∠AEC - ∠ADE, we get: ∠2 = 104° - 36° = 68°. 6. Through the above reasoning, the final answer is: 68°.

垂线; 直角三角形; 邻补角

Exterior Angle Theorem of Triangle

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

<note>In triangle ADE, angle AEC is an exterior angle of the triangle, angles ADE and DAE are the two interior angles that are not adjacent to the exterior angle AEC</note>, according to the Exterior Angle Theorem of Triangle, <note>the exterior angle AEC is equal to the sum of the two non-adjacent interior angles ADE and DAE, that is, angle AEC = angle ADE + angle DAE</note>.

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 ADE</note> is a geometric figure formed by <note>three non-collinear points A, D, E</note> and their connecting line segments <note>AD, DE, AE</note>. <note>Points A, D, E are the three vertices of the triangle</note>, and <note>line segments AD, DE, AE are the three sides of the triangle</note>.

Definition of Exterior Angle of a Polygon

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

The interior angle of <note>polygon AED</note> is <note>∠DAE</note>, and the angle formed by extending the adjacent sides <note>DE and AE</note> of this interior angle is called the exterior angle of the interior angle <note>∠ADE</note>.


Question: As shown in the figure, the cross-section of a certain tunnel is a semicircle with a radius of 3.4m. A truck with a width of 3.2m can just pass through the tunnel. What is the maximum height of the truck including the cargo (in meters)?

Answer: 3m

1. Let the center of the semicircle be O, the diameter be AB, and the rectangle be QWER (starting from the top left corner and taking the vertices of the rectangle counterclockwise). Draw a line through O perpendicular to QR, intersecting the top side of the rectangle at H and the circumference at J.

2. Obviously, OJ is a radius. According to the perpendicular bisector theorem, OJ perpendicularly bisects QR, so QH = HR.

3. Given that the truck's width is 3.2m, QR = 3.2, HR = 3.2/2 = 1.6. Connecting OR, we know OR is a radius, OR = 3.4. Triangle ORH meets the definition of a right triangle, so triangle ORH is a right triangle.

4. In the right triangle ORH, according to the Pythagorean theorem, OR^2 = RH^2 + OH^2. Substituting OR and RH, we get OH = √(3.4^2 - 1.6^2) = 3 meters.

5. Therefore, the maximum height of the truck and cargo together is 3 meters.

圆; 垂线; 直角三角形; 弦

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 ORH, angle OHR is a right angle (90 degrees)</note>, therefore triangle ORH is a right triangle. <note>Side OH and side RH are the legs, and side OR 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.

The original text: <note>线段OH和线段QR相交形成的角∠OHR是90度</note>, therefore according to the definition of perpendicular lines, <note>线段OH和线段QR互相垂直</note>.

Pythagorean Theorem

<note>In the right triangle △ORH, ∠OHR is a right angle (90 degrees)</note>, sides OH and RH are the legs, and side OR is the hypotenuse, so according to the Pythagorean Theorem, <note>OR^2＝RH^2＋OH^2</note>. Given <note>OR=3.4 meters, HR=1.6 meters</note>, use this theorem to find the length of OH.

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.

In <note>circle O</note>, <note>diameter OJ is perpendicular to chord QR</note>, then according to the <note>Perpendicular Diameter Theorem</note>, <note>diameter OJ bisects chord QR</note>, that is, <note>QH=HR</note>.


Question: As shown in the figure, given that the central angle ∠AOB is 110°, then the inscribed angle ∠ACB is equal to ()

Answer: 125°

1. According to the problem statement, it is known that the central angle ∠AOB measures 110°.

2. Let point E be located on arc AB, and connect point E to points A and B, i.e., connect EA and EB.

3. According to the inscribed angle theorem, the inscribed angle on the corresponding major arc AB is half of the central angle ∠AOB, i.e., ∠AEB = 1/2 ∠AOB.

4. Substitute the given condition, we get ∠AEB = 1/2 × 110° = 55°.

5. According to the supplementary angles theorem of cyclic quadrilateral (corollary 3 of the inscribed angle theorem), ∠AEB + ∠ACB = 180°.

6. Therefore, ∠ACB = 180° - ∠AEB = 180° - 55° = 125°.

6. The final answer is 125°.

圆; 圆心角; 圆周角; 弧; 弦

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

Definition of Inscribed Angle

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

In the figure of this problem, in circle 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, angle ACB is an inscribed angle.##In circle O, <note>the vertex C 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, angle APB 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>Points A, B, and P are on the circle</note>, the central angle corresponding to arc AB is <note>∠AOB</note>, and the inscribed angle is <note>∠APB</note>. According to the Inscribed Angle Theorem, <note>∠APB is equal to half of the central angle ∠AOB corresponding to arc AB</note>, i.e., <note>∠APB = 1/2 ∠AOB</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.

In the <note>cyclic quadrilateral ACBP</note>, <note>the vertices ACBP of the quadrilateral all lie on the circle</note>. According to the <note>Diagonal Supplementary Theorem for Cyclic Quadrilateral</note>, the sum of each pair of opposite angles of the quadrilateral ACBP is equal to 180°. Specifically, <note>∠ACB + ∠APB = 180°; ∠PAC + ∠PBC = 180°</note>.


Question: As shown in the figure, line a ∥ b, line c intersects a and b, ∠1 = 55°, then ∠2 = ()

Answer: 55°

1. Given that line a is parallel to line b, and ∠1 = 55°, ##let the vertical angle of ∠2 be ∠3##.

2. ##According to the parallel axiom 2 of parallel lines, corresponding angles are equal##, we get ∠3 = ∠1 = 55°.

3. ##According to the definition of vertical angles##, ∠2 and ∠3 are vertical angles, we get ∠2 = ∠3 = 55°.

4. Through the above reasoning, the final answer is 55°.

Definition of Vertical Angles

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

In the figure of this problem, line c intersects line b at point Y, forming <note>two angles: ∠2 and ∠3</note>. According to the definition of vertical angles, <note>∠2 and ∠3 are vertical angles</note>. Since vertical angles are equal, <note>∠2=∠3</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 a and b</note> are intersected by <note>a third line c</note>, forming the following geometric relationship: <note>corresponding angles: angle 1 and angle 3 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 a and b</note> are intersected by <note>a line c</note>, where <note>angle 1 and angle 3 are on the same side of the intersecting line c and on the same side of the intersected lines a and b</note>, therefore, <note>angle 1 and angle 3 are corresponding angles</note>. <note>Corresponding angles are equal, i.e., angle 1 is equal to angle 3</note>.


Question: What is the degree measure of ∠ABC?

Answer: 75 degrees

1. Given the condition △ABC, where ∠BAC, ∠ABC, and ∠ACB are 4x°, 5x°, and 3x° respectively, according to the triangle angle sum theorem (the sum of the three interior angles of any triangle is 180°), we obtain the equation: 4x + 5x + 3x = 180.

2. Simplify the equation: 4x + 5x + 3x = 12x, thus 12x = 180.

3. Divide both sides of the equation 12x = 180 by 12, we get x = 15.

4. The degree of ∠ABC is 5x°, substituting the value of x we get ∠ABC = 5 × 15 = 75°.

5. Through the above reasoning, the final answer is 75°.

Definition of an Angle

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

<note>Angle BAC is a geometric figure composed of ray AB and ray AC</note>, these two rays have a common <note>endpoint A</note>, this common <note>endpoint A is called the vertex of angle BAC</note>, and ray AB and ray AC are called <note>sides of angle BAC</note>. <note>Angle ABC is a geometric figure composed of ray AB and ray BC</note>, these two rays have a common <note>endpoint B</note>, this common <note>endpoint B is called the vertex of angle ABC</note>, and ray AB and ray BC are called <note>sides of angle ABC</note>. <note>Angle ACB is a geometric figure composed of ray AC and ray BC</note>, these two rays have a common <note>endpoint C</note>, this common <note>endpoint C is called the vertex of angle ACB</note>, and ray AC and ray BC are called <note>sides of angle ACB</note>.

Triangle Angle Sum Theorem

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

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


Question: In rhombus ABCD, AB = 2x + 3 and BC = 5x. Find AD.

Answer: 5

1. Given the rhombus ABCD, ##according to the definition of a rhombus,## side AB and BC are equal, i.e., AB = BC. Given AB = 2x + 3, BC = 5x.

2. According to ##the definition of a rhombus##, all four sides are equal, therefore AB = DC ##= AD## = BC.

3. We can set up the equation 2x + 3 = 5x.

4. Move 2x to the ##right side## of the equation, resulting in 3 = 3x.

5. Solve the equation x = 1.

6. Substitute the value of x into AB = 2x + 3, we get AB = ##2*(1)## + 3 = 5.

7. Since all sides of the rhombus are equal, AD is also equal to 5.

8. Therefore, the length of AD is 5.

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>All sides AB, BC, CD, and DA of rhombus ABCD are equal</note>, therefore <note>AB = BC = CD = DA</note>. Additionally, <note>the diagonals AC and BD of rhombus ABCD bisect each other perpendicularly</note>, that is, <note>the diagonals AC and BD intersect at point E</note>, and <note>∠AED is a right angle (90 degrees)</note>, and <note>AE = EC and BE = ED</note>.


Question: As shown in the figure, AB is the diameter of ⊙O, C and D are points on ⊙O, and AC, BC, CD, OD are connected respectively. If ∠DOB = 140°, then ∠ACD = ()

Answer: 20°

1. Given ∠DOB = 140°. According to the definition of adjacent supplementary angles, ∠AOD = 180° - ∠DOB = 180° - 140° = 40°.

2. According to the definition of central angle, ∠AOD is a central angle.

3. According to the definition of inscribed angle and the inscribed angle theorem, ∠ACD = 1/2 ∠AOD = 1/2 * 40° = 20°.

4. Therefore, the final conclusion is ∠ACD = 20°.

圆; 圆周角; 圆心角; 线段; 弧

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 the points <note>A and B on the circumference</note>, with a length of <note>2 times the radius</note>, that is, <note>AB = 2 * OA</note>.

Definition of Inscribed Angle

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

Definition of Central Angle

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

Inscribed Angle Theorem

<note>In circle O, points A, C, and D are on the circle</note>, <note>the central angle corresponding to arc AD is ∠AOD</note>, <note>the inscribed angle is ∠ACD</note>. According to the Inscribed Angle Theorem, <note>∠ACD is equal to half of the central angle ∠AOD corresponding to the arc AD</note>, that is, <note>∠ACD = 1/2 ∠AOD</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 AOD and angle DOB have a common side OD</note>, <note>their other sides AO and OB are extensions in opposite directions</note>, so <note>angle AOD and DOB are adjacent supplementary angles</note>.


Question: As shown in the figure, in ABC, AB=AC=4cm, BC=6cm, then \cosB=()

Answer: \frac{3}{4}

1. Given triangle ABC, AB = AC = 4cm, BC = 6cm.

2. Draw auxiliary line AD, perpendicular to BC at point D.

3. From the definition of right triangles, triangles ABD and ACD are both right triangles.

4. According to the criteria for congruent right triangles (hypotenuse and one leg), AB = AC, AD = AD, thus right triangles ABD and ACD are congruent.

5. Based on the definition of congruent triangles, BD = 6/2 = 3.

6. In right triangle ABD, using the cosine function, cosB = BD/AC = 4/6 = 3/4.

7. Through the above reasoning, we finally obtain cos∠B = 3 / 4.

等腰三角形; 余弦; 普通三角形; 线段

Definition of Right Triangle

In the figure of this problem, in triangles ABD and ACD, <note>angles ∠ADB and ∠ADC are right angles (90 degrees)</note>, therefore <note>triangles ABD and ACD are right triangles</note>. <note>Sides AD and BD are the legs, side AB is the hypotenuse</note>; <note>sides AD and DC are the legs, side AC is the hypotenuse</note>.

Cosine Function

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

In the figure of this problem, in the right triangle ABD, <note>side BD is the adjacent side of angle B</note>, <note>side AB is the hypotenuse</note>. Therefore, according to the definition of the cosine function, <note>the cosine value of angle B is equal to the ratio of the adjacent side BD to the hypotenuse AB</note>, i.e., cos(B) = <note>BD / AB</note>.

Right Triangle Congruence (Hypotenuse, One Leg)

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

In the figure of this problem, right triangle <note>ABD</note> and right triangle <note>ACD</note>, assuming <note>angle ADB and angle ADC are both right angles (90 degrees)</note>. If <note>side AB is equal to side AC (hypotenuse)</note>, and <note>side AD is equal to side AD (one leg)</note>, then according to the right triangle congruence principle (hypotenuse, one leg), these two right triangles <note>ABD and ACD are congruent</note>.

Definition of Congruent Triangles

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

<note>Triangles ABD and ACD are congruent triangles</note>, the corresponding sides and angles of triangle ABD are equal to those of triangle ACD, namely: <note>side AB = side AC</note> <note>side AD = side AD</note> <note>side BD = side CD</note>, and the corresponding angles are also equal: <note>angle B = angle C</note> <note>angle ADB = angle ADC</note> <note>angle BAD = angle CAD</note>.


Question: As shown in the figure, in the right triangle Rt△ABC, ∠C=90°, AB=10, AC=8, find the value of tanB.

Answer: \frac{4}{3}

1. Given Rt△ABC, ∠ACB=90°, AB=10, AC=8.

2. According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. Let the length of BC be x, then by the Pythagorean theorem: x^2 = AB^2 - AC^2.

3. Substituting the values, we get x^2 = 10^2 - 8^2 = 100 - 64.

4. Calculating, we get x^2 = 36, therefore x = 6.

5. According to the definition of the tangent function, in a right triangle, the tangent of an acute angle is equal to the ratio of the length of the opposite side to the length of the adjacent side. For angle BAC, tan(BAC) = AC/BC.

6. Substituting the values of AC and BC, we get tan(BAC) = 8/6 = 4/3.

7. Through the above reasoning, the final answer is 4/3.

Definition of Right Triangle

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

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

Definition of Tangent Function

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

In the right triangle <note>ABC</note>, <note>angle BAC is an acute angle</note>, <note>side AC is the opposite leg of angle BAC</note>, <note>side BC is the adjacent leg of angle BAC</note>, so <note>the tangent of angle BAC is equal to the length of side AC divided by the length of side BC</note>, that is, <note>tan(BAC) = AC / BC</note>.

Pythagorean Theorem

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

In the diagram of this problem, in the right triangle ABC, <note>angle ACB is a right angle (90 degrees)</note>, <note>sides AC and BC are the legs</note>, <note>side AB is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>AB^2 = AC^2 + BC^2</note>. Given <note>AB = 10</note>, <note>AC = 8</note>, we can use the Pythagorean Theorem to calculate <note>the length of BC</note>, that is <note>BC = √(AB^2 - AC^2) = √(10^2 - 8^2) = √36 = 6</note>.


Question: Circles G, J, and K all intersect at L. If GH = 10, find the measurement. Find JL.

Answer: 5

1. The given condition is that three circles G, J, K intersect at one point L, and GH = 10.

2. According to the figure in the problem, the centers of circles G, J, K are points G, J, and K respectively. G, J, and L are on the same line, and both points H and L are on circle G, and both points G and L are on circle J. According to the definition of radius, GH and LG are radii of circle G, JL is the radius of circle J, GH = LG, GJ = LJ, i.e., LJ = LG/2.

3. Substituting LG = GH = 10, we get JL = 10/2 = 5.

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

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 this problem, <note>circle G has point G as its center</note>, <note>radius is GH</note>; <note>circle J has point J as its center</note>, <note>radius is JL</note>; <note>circle K has point K as its center</note>, <note>radius is KL</note>. All <note>points that are at a distance equal to GH from point G are on circle G</note>, <note>points that are at a distance equal to JL from point J are on circle J</note>, <note>points that are at a distance equal to KL from point K are on circle K</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>GH is the radius of circle G</note>, L is the intersection point of circles G, J, and K, therefore <note>GH and GL are the radii of circle G</note>, <note>JL is the radius of circle J</note>, <note>KL is the radius of circle K</note>.


Question: As shown in the figure, points A, B, C, and D are on circle O, and point E is on the extension of AD. If ∠ABC=60°, then the degree of ∠CDE is ()

Answer: 60°

1. Given quadrilateral ABCD is a cyclic quadrilateral of circle O, ##so according to (Corollary 3 of the Inscribed Angle Theorem) the theorem of supplementary opposite angles in a cyclic quadrilateral##, we get ∠ABC + ∠ADC = 180°.

2. Since point E is on the extension of AD, ∠CDE and ∠ADC are ##adjacent supplementary angles##, according to ##the definition of adjacent supplementary angles##, we get ∠CDE + ∠ADC = 180°.

3. Given ∠ABC = 60°, from the reasoning in step one, we get ∠ADC = 180° - ∠ABC = 180° - 60° = 120°.

4. Substitute the value of ∠ADC into the reasoning in step two, we get ∠CDE + 120° = 180°, thus ∠CDE = 180° - 120° = 60°.

5. Through the above reasoning, we finally get the answer ∠CDE = 60°.

圆; 圆周角; 圆内接四边形; 三角形的外角

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.

In the figure of this problem, the quadrilateral <note>ABCD's four vertices A, B, C, and D are all on the same circle O</note>. This <note>circle is called the circumcircle of quadrilateral ABCD</note>. Therefore, quadrilateral ABCD is a cyclic quadrilateral. According to the properties of cyclic quadrilaterals, it can be concluded that <note>the sum of opposite angles is equal to 180 degrees</note>, that is, <note>∠ABC + ∠ADC = 180 degrees</note>, <note>∠BAD + ∠BCD = 180 degrees</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>In the diagram of this problem, quadrilateral ABCD is a cyclic quadrilateral of circle O, and the vertices A, B, C, D of the quadrilateral are 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°</note>; <note>∠BAD + ∠BCD = 180°</note>.

Definition of Linear Pair of Angles

<note>Angle ADC and angle CDE</note> have a <note>common side DC</note>, and their <note>other sides AD and DE are extensions in opposite directions</note>, so <note>angle ADC and angle CDE are adjacent supplementary angles</note>##.


Question: In the figure, m \angle 1 = 50 and m \angle 3 = 60. Find the measure of \angle 8.

Answer: 120

1. Draw an arrow on each of the lines m and n in the figure to indicate that these two lines are parallel. Draw three arrows on each of the lines p and q to indicate that these two lines are parallel. That is, m∥n, p∥q.

2. Let line n intersect line q at point o. The obtuse angle formed below line q by the intersection of line n and line q is ∠9, and the obtuse angle formed above line q is ∠10. The acute angle formed below line q by the intersection of line n and line q is ∠11, and the acute angle formed above line q is ∠12. According to the parallel postulate 2 of parallel lines and the definition of consecutive interior angles, since m∥n and the parallel lines m and n are cut by line q, ∠3 and ∠9 are consecutive interior angles and are supplementary, that is, ∠3+∠9=180°. Given that ∠3=60°, so ∠9=180°-∠3=180°-60°=120°.

3. According to the definition of vertical angles, the two opposite angles formed by the intersection of line n and line q are equal, so ∠9=∠10, that is, ∠10=120°.

4. According to the parallel postulate 2 of parallel lines and the definition of corresponding angles, since p∥q and the parallel lines p and q are cut by line n, ∠10 and ∠8 are corresponding angles, that is, ∠8=∠10=120°.

5. Through the above reasoning, it is finally concluded that the measure of ∠8 is 120°.

平行线; 内错角; 同位角

Definition of Vertical Angles

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

In the figure of this problem, <note>lines n and q intersect at point k</note>, forming four angles: ∠9, ∠10, ∠11, and ∠12. According to the definition of vertical angles, <note>∠9 and ∠10 are vertical angles</note>, <note>∠11 and ∠12 are vertical angles</note>. Since vertical angles are equal, <note>∠9 = ∠10</note>, <note>∠11 = ∠12</note>.

Parallel Postulate 2 of Parallel Lines

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

<note>In the figure of this problem, two parallel lines m and n are intersected by a third line q, forming the following geometric relationships:</note> 1. Corresponding angles: None. 2. Alternate interior angles: None. 3. Consecutive interior angles: <note>Angle 3 and angle 9 are supplementary, that is, angle 3 + angle 9 = 180 degrees</note>. These relationships indicate that when two parallel lines are intersected by a third line, the consecutive interior angles are supplementary. <note>Two parallel lines p and q are intersected by a third line n, forming the following geometric relationships:</note> 1. Corresponding angles: <note>Angle 8 and angle 10 are equal</note>. 2. Alternate interior angles: None. 3. Consecutive interior angles: None. These relationships indicate that when two parallel lines are intersected by a third line, the corresponding angles are equal.

Definition of Consecutive Interior Angles

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

<note>Two lines m and n</note> are intersected by <note>a third line q</note>, <note>angles ∠3 and ∠9 are on the same side of the intersecting line q</note>, and within <note>the intersected lines m and n</note>, so <note>angles 3 and 9 are consecutive interior angles</note>. <note>Consecutive interior angles ∠3 and ∠9 are supplementary</note>, that is, <note>angle 3 + angle 9 = 180 degrees</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.

In the figure of this problem, <note>two parallel lines p and q</note> are intersected by <note>a line n</note>, where <note>angle 10 and angle 8 are on the same side of the intersecting line n</note>, on the <note>same side of the two lines p and q</note>, therefore <note>angle 10 and angle 8 are corresponding angles</note>. Corresponding angles are equal, that is, <note>angle 10 is equal to angle 8</note>.


Question: The radius of circle O is 13 inches. Radius OB is perpendicular to chord CD, and the length of chord CD is 24 inches. Find OX.

Answer: 5

1. Given that the radius of circle O is 13 inches, radius OB is perpendicular to chord CD, and the length of chord CD is 24 inches.

2. According to the Perpendicular Bisector Theorem, when the radius of a circle is perpendicular to a chord, it bisects the chord. That is, CD is divided into two segments, each of length (m=) CD/2 = 24/2 = 12 inches. Therefore, the lengths of CX and DX are both 12 inches.

3. Since OX is perpendicular to CD, ∠OXC=90°, so according to the definition of a right triangle, triangle OXC is a right triangle. According to the Pythagorean theorem, in right triangle OXC, OC² = XC² + XO² = 144 + XO² = 169.

4. The equation is derived: XO² = 169 - 144 = XO² = 25, XO = √25 = 5.

5. Through the above reasoning, the final answer is: the length of XO is 5 inches.

圆; 垂线; 直角三角形; 弦; 中点

Definition of Radius

<note>In the diagram of this problem, in circle O, point O is the center of the circle, point B is any point on the circle, and segment OB is the segment from the center of the circle to any point on the circle, therefore segment OB is the radius of the circle.</note>

Definition of Chord

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

<note>In circle O</note>, points <note>C and D are any two points on the circle</note>, <note>line segment CD connects these two points</note>, so <note>line segment CD is a chord of circle O</note>.

Perpendicular Diameter Theorem

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

In the figure of this problem, in circle O, <note>radius OB is perpendicular to chord CD</note>, then according to the Perpendicular Diameter Theorem, <note>radius OB bisects chord CD</note>, that is, <note>CX = XD = CD/2 = 12 inches</note>, and <note>radius OB bisects the two arcs subtended by chord CD</note>, that is, <note>arc CB = arc DB</note>.

Pythagorean Theorem

In the figure of this problem, in the right triangle OXC, <note>∠OXC is a right angle (90 degrees)</note>, <note>sides OX and CX are the legs</note>, <note>side OC is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>OC² = OX² + CX²</note>, that is 13² = OX² + 12².

Definition of Right Triangle

<note>Angle OXC is a right angle (90 degrees)</note>, therefore triangle OXC is a right triangle. <note>Side CX and side OX are the legs</note>, <note>side OC is the hypotenuse</note>.


Question: As shown in the figure, in the cyclic quadrilateral ABCD, ∠ABC=120°, then the exterior angle ∠ADE of the quadrilateral ABCD is ()

Answer: 120°

1. Given that quadrilateral ABCD is a cyclic quadrilateral, according to the ##(Inscribed Angle Theorem Corollary 3) Cyclic Quadrilateral Opposite Angles Supplementary Theorem##, we obtain that ∠ADC and ∠ABC are supplementary, i.e., ∠ADC + ∠ABC = 180°.

 2. Given ∠ABC = 120°, then according to the conclusion from step 1, we get ∠ADC = 180° - 120° = 60°.

 3. ##∠ADE and ∠ADC conform to the definition of adjacent supplementary angles, being adjacent supplementary angles##.

 4. From step 2, we get ∠ADC = 60°, so ∠ADE = 180° - 60° = 120°.

 5. Through the above reasoning, we finally obtain the answer ∠ADE = 120°.

圆内接四边形; 圆周角; 三角形的外角; 邻补角; 弧

Cyclic Quadrilateral

<note>The four vertices A, B, C, and D of quadrilateral ABCD are 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, <note>the sum of the opposite angles is equal to 180 degrees</note>, that is, <note>∠ABC + ∠ADC = 180 degrees</note>, <note>∠BAD + ∠BCD = 180 degrees</note>.

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

Definition of Linear Pair of Angles

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

<note>Angle ADE and angle ADC share a common side AD</note>, and their other sides <note>DE and DC are extensions in opposite directions</note>, so <note>angle ADE and angle ADC are adjacent supplementary angles</note>.


Question: Q is the centroid and B E = 9. Find B Q.

Answer: 6

1. Given that Q is the centroid of triangle ABC, and the length from B to E is BE=9. We need to find the length of BQ.

2. According to the centroid theorem, the centroid divides each median into a ratio of 2:1 (the distance from the centroid to the vertex is twice the distance from the centroid to the midpoint of the side).

3. In triangle ABC, BE is the median of side AC, and Q is the centroid of the triangle, so BQ:QE=2:1.

4. Let QE=x, then BQ=2x.

5. According to the definition of line segments, point Q is on the line segment BE, so BQ + QE = BE.

6. Substituting the given data, we get 2x + x = 9.

7. Combining like terms, 3x=9. Thus, we get x=3.

8. Given x=3, we find BQ=2*3=6.

9. Through the above reasoning, the final answer is BQ=6.

点; 线段; 中点; 普通三角形

Centroid Theorem

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

<note>In the figure of this problem, in triangle ABC, point Q is the centroid of the triangle</note>. <note>The three medians of the triangle are line segments AD, BE, and CF, which intersect at point Q</note>. According to the Centroid Theorem, <note>point Q divides each median into two segments in a 2:1 ratio, with the longer segment connecting to the vertex</note>. For example, <note>on line segment BE, point Q divides it into two segments, with the longer segment being BQ (connecting to vertex B) and the shorter segment being QE</note>. Similarly, <note>on line segments AD and CF, point Q also divides them into two segments in a 2:1 ratio</note>.

Definition of Line Segment

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

<note>Line segment BE</note> is a part of a straight line, containing endpoint <note>B</note> and endpoint <note>E</note> and all points in between. <note>Line segment BE</note> has two endpoints, which are <note>E</note> and <note>B</note>, and every point on <note>line segment BE</note> is located between endpoint <note>E</note> and endpoint <note>B</note>.


Question: Find the area of the figure.

Answer: 144

1. Given that the length of the rectangle is 16 inches and the width is 9 inches, according to the definition of a rectangle, the opposite sides are equal, and each angle is a right angle (90 degrees), ##let the four angles of the rectangle be ∠A, ∠B, ∠C, ∠D##.

2. According to the rectangle area formula: Area of the rectangle = length × width.

3. Substituting the given conditions, we get Area = 16 inches × 9 inches = 144 square inches.

4. Through the above reasoning, the final answer is 144 square inches.

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>A quadrilateral is a rectangle</note>, with all its interior angles being right angles (90 degrees), and the length is 16 inches, width is 9 inches. Due to the definition of a rectangle, <note>each interior angle is a right angle (90 degrees), i.e., ∠A = ∠B = ∠C = ∠D = 90 degrees</note>.

Formula for the Area of a Rectangle

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

In this problem diagram, <note>in the rectangle, the side lengths of 16 inches and 9 inches are the length and width of the rectangle</note>, so <note>the area of the rectangle = 16 inches × 9 inches</note>.


Question: As shown in the figure, Xiaoming designed two right angles to measure the width of the river BC. He measured AB=2 meters, BD=\frac{7}{3} meters, CE=9 meters, then the width of the river BC is ()

Answer: \frac{40}{7}米

1. In △ABD and △ACE, it is known that ∠DBA=90°, ∠ECA=90°, and ∠A is shared. According to the theorem of similarity triangles (AA), △ABD is similar to △ACE.

2. According to the definition of similarity triangles, we have AB/AC = BD/CE.

3. Given that AB=2 meters, BD=7/3 meters, and CE=9 meters, substituting into the equation gives: 2/AC = (7/3)/9.

4. Calculation gives: AC = 2 * 9 / (7/3) = 54 / 7.

5. From the figure, we know AC=BC+AB, so BC = AC - AB.

6. Substituting the values gives: BC = 54/7 - 2 = 54/7 - 14/7 = 40/7 meters.

7. Through the above reasoning, the final answer is: BC = 40/7 meters.

直角三角形; 垂线; 平行线

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 ABD and ACE are similar triangles</note>. According to the definition of similar triangles: <note>∠BAD = ∠CAE</note>, <note>∠ABD = ∠ACE</note>, <note>∠ADB = ∠AEC</note>; <note>AB/AC = BD/CE</note>.

Similarity Theorem for Triangles (AA)

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

In the figure of this problem, △ABD and △ACE are known to have <note>∠DBA=∠ECA=90°</note> and <note>angle A is shared</note>, so <note>△ABD is similar to △ACE</note>.


Question: Find y.

Answer: 45

1. This figure is a square with a diagonal of 13, and all four sides are equal in length, i.e., each side length is x. ##Therefore, the diagonal divides the square into two isosceles right triangles##.

2. ##In an isosceles right triangle, according to the triangle sum theorem, the measure of the acute angle can be calculated as = 1/2(180° - 90°) = 45°##.

####

##3##. In summary, the diagonal divides one interior angle of the square into two 45-degree angles, so y is 45 degrees.

正方形; 直角三角形; 三角形的外角; 对称

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°).

Original text: <note>The four sides of the quadrilateral are equal, each being x</note>, and <note>all interior angles are 90 degrees</note>, so the quadrilateral is a square.

Definition of Diagonal

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

<note>In the figure of this problem, in the square</note>, <note>the diagonal</note> is the line segment connecting one vertex to its non-adjacent vertex.

Definition of Right Triangle

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

The <note>diagonal of the square</note> divides the <note>square into two right triangles</note>. Each right triangle has two legs that are <note>the side length x of the square</note>, and the hypotenuse is <note>the diagonal of the square 13</note>. Therefore, these two <note>triangles are both right triangles</note>.

Definition of Isosceles Triangle

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

<note>In the figure of this problem, the two right-angled triangles have equal lengths for their two legs</note>, therefore both triangles are isosceles triangles.

Triangle Angle Sum Theorem

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

<note>In an isosceles right triangle, two acute angles and one right angle are the three interior angles of the triangle</note>, according to the <note>Triangle Angle Sum Theorem</note>, the sum of the three interior angles is <note>equal to 180°</note>.


Question: As shown in the figure, ⊙O is the circumcircle of △ABC, ∠A=70°, then the measure of ∠BOC is ()

Answer: 140°

1. Given that circle O is the circumcircle of △ABC, and from the given conditions, ∠A = 70°.

2. According to the inscribed angle theorem, the inscribed angle is equal to half of the central angle that subtends the same arc, i.e., ∠BOC = 2×∠BAC.

3. From the given conditions, ∠BAC = 70°, substituting into the previous formula gives ∠BOC = 2×70° = 140°.

4. Through the above reasoning, the final answer is 140°.

圆; 圆周角; 圆心角; 弧; 普通三角形

Circumcircle of a Triangle

The circumcircle of a triangle is a circle that passes through all three vertices of the triangle.

<note>The three vertices of triangle ABC are points A, B, and C</note>. A circle can be drawn through these three vertices, which is called the <note>circumcircle of triangle ABC</note>. <note>The center O is the center of the circumcircle of triangle ABC</note>, and <note>points A, B, and C are all on this circle</note>. Therefore, <note>the radius of circle O is equal to OA, OB, and OC, that is, OA=OB=OC</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 circle O, <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>.

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

Inscribed Angle Theorem

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

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


Question: As shown in the figure, AB is the diameter of ⊙O, and points C and D are on ⊙O. If ∠BOD=130°, then the degree of ∠ACD is ()

Answer: 25°

1. Given that point A and point B are the endpoints of the diameter of ⊙O, ##according to the definition of a straight angle, ∠AOB=180°, ∠AOB=∠BOD+∠AOD.##

2. ##Given ∠BOD=130°, we get ∠AOD=∠AOB-∠BOD=180°-∠BOD=50°.##

3. ##According to the definition of an inscribed angle, ∠ACD is the inscribed angle of circle O; according to the definition of a central angle, ∠AOD is the central angle of circle O. Then, according to the inscribed angle theorem, ∠ACD is equal to half of the corresponding central angle ∠AOD, so ∠ACD=1/2∠AOD.##

4. Substituting ∠AOD=50°, we get ∠ACD=1/2×50°=25°.

5. Through the above reasoning, the final answer is 25°.

Definition of Central Angle

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. The angle <note>∠AOB</note> formed by the lines <note>OA</note> and <note>OB</note> is called a central angle; similarly, the angle <note>∠BOD</note> formed by the lines <note>OB</note> and <note>OD</note> is also called a central angle.

Definition of Inscribed Angle

In the figure of this problem, in circle O, the vertex C of angle ACD is on the circumference, and the two sides of angle ACD intersect circle O at points A and D respectively. Therefore, angle ACD is an inscribed angle.

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>Ray OA</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 <note>definition of straight angle</note>, the <note>measure of a straight angle is 180 degrees</note>, i.e., <note>angle AOB = 180 degrees</note>.

Inscribed Angle Theorem

In the figure of this problem, in circle O, <note>points A, B, C, D are on the circle</note>, <note>the central angles corresponding to arc AD and arc AC are ∠AOD</note>, <note>the inscribed angle is ∠ACD</note>. According to the Inscribed Angle Theorem, <note>∠ACD is equal to half of the central angle ∠AOD corresponding to arc AD</note>, that is, <note>∠ACD = 1/2 ∠AOD</note>.


Question: Find P N.

Answer: 30

1. From the figure, it is known that ∠LMP = 25° and ∠NMP = 25°, and ∠MLP = 90° and ∠MNP = 90°.

2. According to the congruent triangles determination theorem (AAS), it is known that ∠LMP = ∠NMP = 25°, ∠MLP = ∠MNP = 90°, and the opposite sides of ∠MLP and ∠MNP in the two triangles are both MP, so triangle MPL ≌ triangle MPN.

3. By the definition of congruent triangles, the corresponding sides of the two triangles are equal, so LP = NP, i.e., 3x + 6 = 4x - 2.

4. Solve the equation: 6 + 2 = 4x - 3x, and finally solve: x = 8.

5. Substitute the obtained value of x into: 4x - 2 to find PN, and solve: PN = 4 * 8 - 2 = 30.

6. After the above reasoning, the final answer is that the length of PN is 30.

直角三角形; 内错角; 同位角; 垂线

Definition of Triangle

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

Original: <note>三角形MLP是由三个非共线点M、L、P及其连接线段ML、LP、MP组成的几何图形</note>。点<note>M、L、P分别是三角形的三个顶点</note>，线段<note>ML、LP、MP分别是三角形的三条边</note>。<note>三角形MNP是由三个非共线点M、N、P及其连接线段MN、NP、MP组成的几何图形</note>。点<note>M、N、P分别是三角形的三个顶点</note>，线段<note>MN、NP、MP分别是三角形的三条边</note>。

Translation: <note>The triangle MLP is a geometric figure formed by three non-collinear points M, L, P and their connecting line segments ML, LP, MP</note>. The points <note>M, L, P are the three vertices of the triangle</note>, and the line segments <note>ML, LP, MP are the three sides of the triangle</note>. <note>The triangle MNP is a geometric figure formed by three non-collinear points M, N, P and their connecting line segments MN, NP, MP</note>. The points <note>M, N, P are the three vertices of the triangle</

Congruence Theorem for Triangles (AAS)

Two triangles are congruent if two angles and the included side of one triangle are respectively equal to two angles and the included side of the other triangle.

<note>Angle LMP is equal to angle NMP</note>, <note>angle MLP is equal to angle MNP</note>, and <note>side MP is equal to MP</note>. Since two angles and the side opposite one of these angles are equal in both triangles, according to the Congruence Theorem for Triangles (AAS), we can conclude that <note>triangle MPL is congruent to triangle MPN</note>.

Definition of Congruent Triangles

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

<note>Triangle MPL and triangle MPN are congruent triangles</note>, <note>The corresponding sides and angles of triangle MPL are equal to those of triangle MPN</note>, namely:

<note>Side ML</note> = <note>Side MN</note>
<note>Side MP</note> = <note>Side MP</note>
<note>Side PL</note> = <note>Side PN</note>

At the same time, the corresponding angles are also equal:
<note>Angle LMP</note> = <note>Angle NMP</note>
<note>Angle MLP</note> = <note>Angle MNP</note>
<note>Angle MPL</note> = <note>Angle MPN</note>


Question: As shown in the figure, quadrilateral ABCD is an inscribed quadrilateral of ⊙O. If ∠C=140°, then the measure of ∠BOD is ()

Answer: 80°

1. Given that quadrilateral ABCD is a cyclic quadrilateral of ⊙O, therefore according to the ##(Corollary 3 of the Inscribed Angle Theorem) the theorem of supplementary opposite angles in a cyclic quadrilateral, in a cyclic quadrilateral, the sum of each pair of opposite angles is equal to 180°##, we get ∠C + ∠A = 180°.

2. From the given condition ∠C = 140°, substituting into the conclusion from the previous step, we get ##∠A = 180° - ∠C = 180° - 140° = 40°##.

3. According to the Inscribed Angle Theorem, the central angle is ##equal to twice the inscribed angle corresponding to the arc it subtends##. In this problem, the central angle ∠BOD corresponds to the inscribed angle ∠BAD.

4. Therefore, the central angle ∠BOD = 2 × ∠BAD = 2 × ∠A.

5. Substituting the previously calculated ∠A = 40°, we get ∠BOD = 2 × 40° = 80°.

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

圆内接四边形; 圆心角; 圆周角; 圆

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>quadrilateral ABCD</note> has <note>four vertices A, B, C, and D all lying 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>.

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.

In the figure of this problem, quadrilateral ABCD is a cyclic quadrilateral, <note>the vertices A, B, C, and D of the quadrilateral are on the circle.</note> According to the Diagonal Supplementary Theorem for Cyclic Quadrilateral, the sum of each pair of opposite angles of quadrilateral ABCD is equal to 180°. Specifically, ∠C + ∠A = 180°, ∠B + ∠D = 180°. The problem states that <note>∠C = 140°</note>, therefore <note>∠A = 180° - ∠C</note> = <note>180° - 140° = 40°.</note>

Inscribed Angle Theorem

In the problem diagram, in circle O, points A, B, C, and D are on the circle, the central angle corresponding to arc BD is ∠BOD, and the inscribed angle is ∠BAD. According to the Inscribed Angle Theorem, ∠BAD is equal to half of the central angle ∠BOD corresponding to arc BD, that is, ∠BAD = 1/2 ∠BOD.

Definition of Central Angle

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

Definition of Inscribed Angle

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


Question: Find the value of x in the figure below, accurate to one decimal place?

Answer: 22.5

1. From the labels in the figure, it is known that there is a right angle. According to the definition of a right triangle, this triangle is a right triangle. Given that one leg is 12 and the hypotenuse is 25.5, we need to find the length of the other leg x.

2. According to the Pythagorean theorem, for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs, i.e., a^2 + b^2 = c^2.

3. Let 12 be leg a, x be leg b, and 25.5 be the hypotenuse c. Substituting into the Pythagorean theorem, we get 12^2 + x^2 = 25.5^2.

4. Calculate 12^2 and 25.5^2, obtaining 12^2 = 144 and 25.5^2 = 650.25.

5. Substitute the results into the equation, obtaining 144 + x^2 = 650.25.

6. Solve this equation by first rearranging terms to get x^2 = 650.25 - 144.

7. Calculate to get x^2 = 506.25.

8. Take the square root of 506.25 to get x = √506.25 = 22.5.

9. Therefore, the length of the leg x is approximately 22.5.

Definition of Right Triangle

<note>This triangle is a right triangle</note>, <note>one of the interior angles is 90 degrees</note>. Given that <note>one leg is 12</note>, <note>the other leg is x</note>, <note>and the hypotenuse is 25.5</note>.

Definition of Triangle

In the figure of this problem, let the triangle be <note>triangle ABC</note>, <note>triangle ABC</note> is a geometric figure composed of <note>three non-collinear points A, B, C</note> and their connecting line segments <note>AC, AB, BC</note>. <note>Points A, B, C are the three vertices of the triangle</note>, <note>line segments AC, AB, BC are the three sides of the triangle</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 a right triangle, the legs are 12 and x respectively, and the hypotenuse is 25.5</note>, according to the Pythagorean Theorem, <note>the square of the hypotenuse is equal to the sum of the squares of the two legs</note>, i.e., <note>25.5^2 = 12^2 + x^2</note>.


Question: As shown in the figure, AB and CD are two diameters of ⊙O, and chord DE∥AB. If arc DE is an arc of 40°, then ∠BOC=()

Answer: 110°

1. Connect OE. According to the problem statement, we know that arc DE is an arc of 40°.

2. According to the definition of the central angle, we get ∠DOE=40°.

3. Since OD=OE, according to the definition of an isosceles triangle, △ODE is an isosceles triangle. Based on the properties of an isosceles triangle, we get ∠ODE=(180°-∠DOE)/2=(180°-40°)/2=70°.

4. Since chord DE is parallel to AB, according to the parallel postulate 2 and the definition of corresponding angles, we get ∠AOC=∠ODE=70°.

5. According to the definition of supplementary angles, ∠BOC=180°-∠AOC, we get ∠BOC=180°-70°=110°.

6. Through the above reasoning, we finally get the answer ∠BOC=110°.

圆; 圆心角; 弧; 弦; 平行线

Definition of Central Angle

In <note>circle O</note>, <note>point E</note> and <note>point D</note> are two points on the circle, and <note>the center of the circle is point O</note>. The angle formed by the lines <note>OD</note> and <note>OE</note>, <note>∠EOD</note>, is called the central angle.

Definition of Arc

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

<note>In the diagram of this problem, there are two points D and E on circle O</note>, arc DE is the segment of the curve connecting these two points. According to the definition of arc, <note>arc DE is the segment of the curve between two points D and E on the circle</note>, and <note>the problem states its degree measure is 40°</note>.

Definition of Isosceles Triangle

<note>Side OD and side OE are equal</note>, therefore <note>triangle ODE is an isosceles triangle</note>.

Definition of Corresponding Angles

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

<note>Two parallel lines AB and DE are intersected by a line CD</note>, where <note>angle EDO and angle AOC are on the same side of the intersecting line CD</note>, <note>on the same side of the intersected lines AB and ED</note>, thus <note>angle EDO and angle AOC are corresponding angles</note>. Corresponding angles are equal, that is <note>angle EDO is equal to angle AOC</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 figure of this problem, in the isosceles triangle ODE, side OD and side OE are equal</note>. Therefore, according to the properties of isosceles triangles, the angles opposite the equal sides are equal, that is, <note>∠ODE = ∠OED</note>.

Parallel Postulate 2 of Parallel Lines

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

In the figure of this problem, <note>two parallel lines DE and AB are intersected by the line OD</note>, forming the following geometric relationships: ##1. <note>Corresponding angles: ∠ODE and ∠AOC are equal</note>. 2. Alternate interior angles are equal. 3. Consecutive interior angles are supplementary.## 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.##

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 COA and angle COB share a common side CO</note>, <note>their other sides AO and OB are extensions of each other in opposite directions</note>, so <note>angle AOC and angle COB are adjacent supplementary angles</note>.

Triangle Angle Sum Theorem

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

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


Question: In the figure, m \angle 9 = 75. Find the measure of \angle 6.

Answer: 105

1. Given ∠6 and ∠9 are same-side interior angles, since line n and line m are parallel, line t is their transversal. By Parallel Postulate 2 of parallel lines, same-side interior angles are supplementary, so ∠6 + ∠9 = 180°.

2. According to the problem statement, ∠9 = 75°, substitute into the above equation: ∠6 + 75° = 180°, thus ∠6 = 105°.

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3. Through the above reasoning, the final answer is 105°.

对顶角; 内错角; 平行线

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

In the figure of this problem, <note>line m and line n are intersected by a third line t</note>, <note>∠6 and ∠9 are on the same side of the transversal t</note>, <note>and within the intersected lines m and n</note>, so <note>∠6 and ∠9 are consecutive interior angles</note>. <note>Consecutive interior angles ∠6 and ∠9 are supplementary</note>, that is, <note>∠6 + ∠9 = 180°</note>.

Parallel Postulate 2 of Parallel Lines

Original: <note>In the figure of this problem, two parallel lines m and n are intersected by a third line t</note>, forming the following geometric relationship: <note>Consecutive interior angles: angle 6 and angle 9 are supplementary</note>, that is, <note>angle 6 + angle 9 = 180 degrees</note>. This relationship indicates that when two parallel lines are intersected by a third line, <note>consecutive interior angles are supplementary</note>.


Question: An airplane flies from Des Moines to Phoenix, then to Atlanta, and finally back to Des Moines, as shown in the figure. If the total distance traveled is 3482 miles, find the distance from Phoenix to Atlanta (unit: miles).

Answer: 1591

1. From the figure and description in the problem, we can deduce that the total sum of the three paths is 3482 miles, where the distance from Des Moines to Phoenix is 110x + 53 miles, the distance from Phoenix to Atlanta is 150x + 91 miles, and the distance from Atlanta to Des Moines is 73.8x miles.

2. According to the problem, the total distance is 3482 miles, i.e., (110x + 53) + (150x + 91) + 73.8x = 3482.

3. Expanding and simplifying the above equation, we get 110x + 150x + 73.8x + 53 + 91 = 3482.

4. Combining like terms, we get 333.8x + 144 = 3482.

5. Simplifying the equation, first subtract 144 from both sides, we get 333.8x = 3338.

6. Then divide both sides by 333.8, we get x = 10.

7. Substitute x = 10 into the distance formula from Phoenix to Atlanta, 150x + 91.

8. Calculating, the distance from Phoenix to Atlanta is 150 * 10 + 91 = 1591 miles.

9. Through the above reasoning, the final answer is 1591.

Definition of Triangle

Original text: <note>A triangle is formed by three non-collinear points Des Moines, Phoenix, and Atlanta</note> and their connecting <note>line segment from Des Moines to Phoenix (110x + 53)</note>, <note>line segment from Phoenix to Atlanta (150x + 91)</note>, <note>line segment from Atlanta to Des Moines (73.8x)</note>. <note>Points Des Moines, Phoenix, and Atlanta are the three vertices of the triangle</note>, <note>and the line segments from Des Moines to Phoenix, Phoenix to Atlanta, and Atlanta to Des Moines are the three sides of the triangle</note>.

Formula for the Perimeter of a Triangle

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

<note>In the figure of this problem, it is known that the three sides of the triangle are 110x+53, 73.8x, and 150x+91 respectively</note>. According to the formula for the perimeter of a triangle, the perimeter of the triangle is equal to the sum of the lengths of its three sides, that is, <note>Perimeter P = (110x + 53) + (150x + 91) + 73.8x</note>.


Question: If AB = 12, AC = 16, and ED = 5, find AE.

Answer: 15

1. Given conditions are AB = 12, AC = 16, ED = 5.

2. Let BC = x. Since AC = BC + AB, we can deduce x + 12 = 16, therefore x = 4.

3. According to the theorem of parallel lines dividing segments proportionally, given that in △ADC, BE is parallel to CD, thus AB/BC = AE/ED. Substituting the corresponding values gives AE / 5 = 12 / 4.

4. Solve this proportion equation: AE / 5 = 12 / 4. Therefore AE = 5 * (12 / 4) = 15.

5. Through the above reasoning, the final answer is AE = 15.

普通三角形; 内错角; 平移

Proportional Segments Theorem

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

<note>In the figure of this problem, in triangle ADC, line BE is parallel to side CD and intersects the other two sides AC and AD (or their extensions) at points B and E</note>, then according to the Proportional Segments Theorem, we have: <note>AB/BC = AE/ED</note>, that is, the intercepted segments are proportional to the corresponding segments of the original triangle.

Definition of Triangle

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

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


Question: As shown in the figure, AB∥CD, ∠B=20°, ∠D=60°, then the degree of ∠BED is ()

Answer: 80°

2. Through point E, draw EF∥AB. ##According to the transitivity of parallel lines,## we get CD∥EF.

3. ##According to Parallel Postulate 2 and the definition of alternate interior angles,## alternate interior angles are equal, thus ∠BEF=∠B=20°.

4. ##According to Parallel Postulate 2 and the definition of alternate interior angles,## alternate interior angles are equal, thus ∠DEF=∠D=60°.

5. ##Since ∠BED is divided by the parallel line EF into ∠BEF and ∠DEF,## we get ∠BED=∠BEF+∠DEF=20°+60°=80°.

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

平行线; 内错角; 同旁内角

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

Parallel Postulate 2 of Parallel Lines

In the diagram of this problem, <note>two parallel lines AB and EF are intersected by a third line BE</note>, forming the following geometric relationships: 1. Corresponding angles: none. 2. Alternate interior angles: <note>angle B and angle BEF 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>. <note>Two parallel lines CD and EF are intersected by a third line DE</note>, forming the following geometric relationships: 1. Corresponding angles: none. 2. Alternate interior angles: <note>angle D and angle DEF 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>.

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>EF</note> are intersected by a line <note>BE</note>, where <note>angle B</note> and <note>angle BEF</note> are located between the two parallel lines and on opposite sides of the intersecting line <note>BE</note>. Therefore, <note>angle B</note> and <note>angle BEF</note> are alternate interior angles. Alternate interior angles are equal, i.e., <note>angle B equals angle BEF</note>. Two parallel lines <note>CD</note> and <note>EF</note> are intersected by a line <note>DE</note>, where <note>angle D</note> and <note>angle DEF</note> are located between the two parallel lines and on opposite sides of the intersecting line <note>DE</note>. Therefore, <note>angle D</note> and <note>angle DEF</note> are alternate interior angles. Alternate interior angles are equal, i.e., <note>angle D equals angle DEF</note>.

Transitivity of Parallel Lines

If two lines are each parallel to a third line, then those two lines are parallel to each other.

In the figure of this problem, <note>line EF and line CD</note> are respectively parallel to <note>line AB</note>. According to the transitivity of parallel lines, if <note>line EF is parallel to line AB</note>, and <note>line CD is also parallel to line AB</note>, then <note>line EF and line CD are mutually parallel</note>. Therefore, <note>line EF is parallel to line CD</note>.


Question: Quadrilateral D E F G is a rectangle. If m \angle E D F = 5 x - 3 and m \angle D F G = 3 x + 7, find m \angle E D F.

Answer: 22

1. Given quadrilateral DEFG is a rectangle, according to ##Parallel Line Axiom 2, alternate interior angles are equal##, therefore ##∠EDF = ∠DFG##.

2. According to the given conditions, ##the degree of ∠EDF## = 5x - 3 and ##the degree of ∠DFG## = 3x + 7.

3. Since ##∠EDF = ∠DFG##, therefore, 5x - 3 = 3x + 7.

4. By solving the equation 5x - 3 = 3x + 7, moving 3x to the left side and the constant term to the right side gives:

   5x - 3x = 7 + 3.

5. Combining like terms, we get 2x = 10.

6. Dividing both sides of 2x = 10 by 2, we get x = 5.

7. Substituting x = 5 into ##the degree of ∠EDF## = 5x - 3, we get ##the degree of ∠EDF## = 5 * 5 - 3.

8. The calculation result is ##the degree of ∠EDF## = 25 - 3 = 22.

9. Thus, the final result is ##the degree of ∠EDF## = 22.

矩形; 邻补角; 对顶角

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>Quadrilateral D E F G is a rectangle</note>, with interior angles <note>∠DGF</note>, <note>∠DEF</note>, <note>∠EFG</note>, <note>∠EDG</note> all being right angles (90 degrees), and sides <note>DE</note> and <note>FG</note> are parallel and equal in length, sides <note>DG</note> and <note>EF</note> are parallel and equal in length.

Parallel Postulate 2 of Parallel Lines

<note>Two parallel lines DE and GF</note> are intersected by <note>a third line DF</note>, forming the following geometric relationship: alternate interior angles: <note>angle EDF and angle DFG are equal</note>. This relationship indicates that when two parallel lines are intersected by a third line, the alternate interior angles are equal.

Definition of Alternate Interior Angles

In the diagram of this problem, <note>two parallel lines DE and GF are intersected by a line DF</note>, where <note>angle EDF and angle DFG are between the two parallel lines and on opposite sides of the intersecting line DF</note>, thus <note>angle EDF and angle DFG are alternate interior angles</note>. Alternate interior angles are equal, that is, <note>angle EDF is equal to angle DFG</note>.


Question: For a pair of similar figures, find the area of the green figure.

Answer: 9

1. Given two quadrilaterals are similar, the diagonals of the two quadrilaterals are 8yd and 4yd respectively, and the ratio of the diagonals is 8:4, which is 2:1.

2. According to the theorem that the ratio of the areas of similar polygons is equal to the square of the similarity ratio, if the ratio of the side lengths of two similar polygons is k, then the ratio of their areas is equal to k squared. Given that the ratio of the diagonals of the two quadrilaterals is 2:1, the ratio of the areas of the two quadrilaterals is 2:1^2, which is 4:1.

3. Given the area of the blue quadrilateral A=36yd^2, the area of the green quadrilateral is A/4, which is 36/4=9yd^2.

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>In the figure of this problem</note>, let <note>the blue quadrilateral be ABCD</note>, and <note>the green quadrilateral be EFGH</note>. In the figure of this problem, <note>the quadrilateral ABCD is composed of four line segments AB, BC, CD, and DA</note>, these line segments are called <note>the sides of the quadrilateral</note>. <note>The quadrilateral ABCD has four vertices, which are point A, point B, point C, and point D</note>, and it has <note>four interior angles, which are angle ABC, angle BCD, angle CDA, and angle DAB</note>. <note>The quadrilateral EFGH is composed of four line segments EF, FG, GH, and HE</note>, these line segments are called <note>the sides of the quadrilateral</note>. <note>The quadrilateral EFGH has four vertices, which are point E, point F, point G, and point H</note>, and it has <note>four interior angles, which are angle EFG, angle FGH, angle GHE, and angle HEF</note>.

Area Ratio Theorem of Similar Polygons

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

<note>Polygon ABCD and polygon EFGH are similar figures</note>, <note>the ratio of side BD to side FH is 2</note>, that is, the similarity ratio is 2. Therefore, <note>the ratio of the area of figure ABCD to the area of figure EFGH is equal to the square of 2</note>. That is: <note>area ratio = square of 2</note>.


Question: Find S T.

Answer: 19

1. Given that △SRT is an isosceles triangle, SR=4x-4, RT=3x+3. According to the properties of an isosceles triangle, SR=RT, so we have the equation 4x-4 = 3x+3.

2. Solving the equation 4x-4 = 3x+3, first move all x to one side of the equation, resulting in x - 4 = 3.

3. Continue solving the equation to get x = 7.

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##4##. Now, according to the problem, find the length of ST, given ST = 12 + x, substitute x = 7 to get ST = 12 + 7 = 19.

##5##. Through the above reasoning, the final answer is ST = 19.

Definition of Isosceles Triangle

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

<note>Side SR and side RT are equal</note>, therefore <note>triangle SRT is an isosceles triangle</note>.

Properties of Isosceles Triangle

<note>In the figure of this problem, in the isosceles triangle SRT, side SR and side RT are equal</note>. Therefore, according to the properties of isosceles triangles, the angles opposite the equal sides are equal, that is, <note>∠RST = ∠RTS</note>.