Question: Find m \angle 6.

Answer: 35

##1. In a triangle containing acute angles 5 and 35°, according to the triangle angle sum theorem (the sum of the interior angles of a triangle is 180 degrees), we get angle 5 = 180° - 35° - 90° = 55°.##

##2. Since angle 5 + angle 6 = 90°, we have angle 6 = 90° - angle 5.##

####

##3. From step 1, we know angle 5 = 55°, so angle 6 = 90° - 55° = 35°.##

##4##. Through the above reasoning, the final answer is angle 6 = 35°.

直角三角形; 三角形的外角

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.

In the figure of this problem, for example, <note>∠1 is a geometric figure formed by two rays that have a common endpoint</note>. This <note>common endpoint is called the vertex of angle 1</note>, and the <note>rays are called the sides of angle 1</note>. Similarly, <note>∠3, ∠2, ∠4, ∠5, ∠6, and other angles are also formed by corresponding rays</note>, and are all expressed in degrees.

Triangle Angle Sum Theorem

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

<note>##In the figure of this problem, the triangle has three interior angles: a 90° angle, a 35° angle, and angle 5</note>, according to the Triangle Angle Sum Theorem, <note>90° + 35° + angle 5 = 180°</note>.####


Question: As shown in the figure, given that the inscribed angle ∠A=50°, then the measure of ∠OBC is ()

Answer: 40°

1. Given ∠A = 50°, according to the inscribed angle theorem, we get ∠BOC = 2∠A = 100°. According to the definition of the central angle, ∠BOC is the central angle of circle O.

2. According to the definition of a circle, OB = OC. In △BOC, according to the definition of an isosceles triangle, △BOC is an isosceles triangle. According to the properties of an isosceles triangle, the base angles opposite the equal sides are equal, thus ∠OBC = ∠OCB.

3. According to the triangle angle sum theorem, the sum of the three interior angles of a triangle is 180°, so in △BOC we have ∠BOC + ∠OBC + ∠OCB = 180°.

4. Substituting the given conditions and the conclusion from the previous step, we get 100° + 2∠OBC = 180°.

5. Solving the equation, we get 2∠OBC = 80°, which means ∠OBC = 40°.

6. Through the above reasoning, we finally get the answer as 40°.

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

Definition of Radius

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

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

Definition of Circle

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

<note>In circle O, point O is the center</note>, <note>the radii are OB and OC</note>. All points in the figure that are <note>equidistant from point O as OB and OC</note> lie on <note>circle O</note>.

Inscribed Angle Theorem

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

In <note>circle O</note>, <note>points A, B, and C are on the circle</note>, <note>arc BC</note> and <note>arc BAC correspond to the central angle ∠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>.

Definition of Isosceles Triangle

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

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

Properties of Isosceles Triangle

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

<note>In isosceles triangle BOC, sides OB and OC are equal</note>. Therefore, according to the properties of isosceles triangles, the angles opposite the equal sides are equal, that is, <note>∠OBC = ∠OCB</note>.

Triangle Angle Sum Theorem

In <note>triangle BOC</note>, <note>angle BOC, angle OBC, and angle OCB</note> are <note>the three interior angles of triangle BOC</note>. According to the Triangle Angle Sum Theorem, <note>angle BOC + angle OBC + angle OCB = 180°</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 of angle BAC (point A) is on the circumference</note>, <note>the two sides of angle BAC intersect circle O at points C and B respectively</note>. Therefore, <note>angle BAC is an inscribed angle</note>.


Question: As shown in the figure, in ⊙O, AB is the diameter, CD is a chord, AB ⊥ CD, and the foot of the perpendicular is point E. Connect CO and AD. If ∠BOC = 30°, then the measure of ∠BAD is ()

Answer: 15°

1. Given AB is the diameter and AB is perpendicular to CD, with the foot of the perpendicular at point E, ##according to the perpendicular diameter theorem, the diameter AB is perpendicular to the chord CD, then the diameter AB bisects the arcs CB and DB corresponding to the chord CD.##, we get BC=BD and CE=DE.

2. Connect CO, based on the given condition ∠BOC=30°.

####

##3. According to the property of the central angle, the arc BC is equal to the degree of its corresponding central angle, that is, arc BC=∠BOC=30°. From the above steps, we can conclude that AB bisects the arc CB, thus arc DB=arc CB=30°##

##4. Given arc BD=arc CB=30°, then the inscribed angle BAD corresponding to the arc BD is equal to half of the central angle corresponding to the arc CB, that is: ∠BAD=1/2∠BOC.##

####

##5. Calculation formula: ∠BAD = (1/2) ∠BOC = (1/2) × 30° = 15°.##

##6##. Through the above reasoning, the final answer is 15°.

圆; 直角三角形; 垂线; 圆周角; 圆心角

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

Definition of Central Angle

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

Perpendicular Diameter Theorem

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

<note>In circle O</note>, <note>diameter AB is perpendicular to chord CD</note>, then according to the Perpendicular Diameter Theorem, <note>diameter AB bisects chord CD</note>, that is, <note>CE=DE</note>, and <note>diameter AB bisects the arcs subtended by chord CD</note>, that is, <note>arc CB=arc BD</note>.

Properties of Central Angles

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

<note>The central angle ∠BOC corresponds to the arc BC</note>. According to the properties of central angles, <note>the degree measure of a central angle is equal to the degree measure of its corresponding arc</note>, that is, <note>the degree measure of ∠BOC = the degree measure of arc BC</note>.

Inscribed Angle Theorem

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


Question: Find K J, if G J = 8, G H = 12, H I = 4.

Answer: 2

1. According to the arrows marked in the figure: KH ∥ IJ.

2. As shown in the figure, line segment KH is inside triangle GJI, and KH ∥ IJ intersects sides GJ and GI at points K and H. According to the theorem of parallel lines dividing segments proportionally, the segments GK and GH intercepted by KH on sides GJ and GI are proportional to the corresponding sides of triangle GJI.

3. The equation obtained is: GK/GJ = GH/GI.

4. Given GH = 12, HI = 4, and GJ = 8, where line segments GH and HI form line segment GI, i.e., GI = GH + HI = 12 + 4 = 16.

5. Substitute the specific values into the proportion equation: GK / 8 = 12 / 16.

6. Cross-multiplying gives: 16GK = 8 * 12.

8. As shown in the figure, GJ is composed of GK and KJ, i.e., GJ = GK + KJ, so KJ = GJ - GK.

10. Substitute GK = 6, then KJ = GJ - GK = 8 - 6 = 2.

12. Through the above reasoning, the final answer is 2.

线段; 等腰三角形; 平行线

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.

<note>Triangle GJI</note> is a geometric figure composed of <note>three non-collinear points G, J, I</note> and <note>their connecting line segments GJ, GI, JI</note>. <note>Points G, J, I</note> are the <note>three vertices</note> of the triangle, and <note>line segments GJ, GI, JI</note> are the triangle's <note>three sides</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 KH and line JI are located in the same plane</note>, and <note>they do not intersect</note>, therefore according to the definition of parallel lines, <note>line KH and line JI are parallel lines</note>.

Proportional Segments Theorem

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

<note>In triangle GJI</note>, <note>line KH is parallel to side IJ</note>, and it intersects the other two sides <note>GJ and GI at points K and H</note>, then according to the Proportional Segments Theorem, we have: <note>GH/GI = GK/GJ</note>. That is, 12/16 = GK/8.


Question: Find m \angle A.

Answer: 60

1. According to the theorem of the sum of the interior angles of a quadrilateral: In any quadrilateral, the sum of the four interior angles is 360°, i.e., ∠A + ∠B + ∠C + ∠D = 360°.

2. Substitute the given angle expressions, we get x° + 2x° + 2x° + x° = 360°.

3. Simplify the equation, we get 6x° = 360°.

4. Divide both sides of the equation by 6, we get x = 60°.

5. Since ∠A = x°, substitute the value of x to get ∠A = 60°.

6. Through the above reasoning, the final answer is 60°.

梯形; 三角形的外角; 同旁内角

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

In the figure of this problem, in <note>quadrilateral ABCD</note>, <note>angle A</note>, <note>angle B</note>, <note>angle C</note>, and <note>angle D</note> are the four interior angles of the quadrilateral. According to the Sum of Interior Angles of a Quadrilateral Theorem, the sum of these four interior angles is 360°, that is, <note>∠A + ∠B + ∠C + ∠D = 360°</note>.


Question: As shown in the figure, in Rt△ABC, ∠ACB=90°, AC=4, BC=3, rotating △ABC around the line where AC is located for one full revolution forms a solid of revolution. Then the lateral surface area of this solid is ()

Answer: 15π

1. Given in the right triangle ABC, ∠ACB=90°, AC=4, BC=3.

2. According to the Pythagorean theorem, AB = √(AC^2 + BC^2) = √(4^2 + 3^2) = √(16 + 9) = √25 = 5.

3. The right triangle ABC rotates around the line where AC lies to form a solid of revolution, which is a cone.

4. The right side BC of the triangle is the radius of the base circumference of the cone, so r=3.

5. Let the slant height of the solid of revolution be AB, then the slant height is 5.

6. According to the surface area formula of the cone, S = π * base radius * slant height. Therefore, S = π * 3 * 5 = 15π.

7. Through the above reasoning, the final answer is 15π.

直角三角形; 旋转; 圆锥; 圆; 圆柱

Definition of Right Triangle

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

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

Pythagorean Theorem

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

<note>In the figure of this problem, in the right triangle ABC, angle ACB is a right angle (90 degrees)</note>, <note>sides AC and BC are the legs, and side AB is the hypotenuse</note>, so according to the Pythagorean Theorem, ##AB^2 = AC^2 + BC^2, i.e., AB = √(AC^2 + BC^2) = √(4^2 + 3^2)## = √(16 + 9) = √25 = 5.

Cone

A cone is a geometric figure with a circular base and a single vertex. Its surface consists of a curved lateral surface extending from the base to the vertex.

In the figure of this problem, <note>the cone is the geometric body obtained by rotating the right triangle ABC around the line where AC is located for one full turn</note>. <note>The base of the cone is a circle</note>, <note>the radius of the circle is BC, which is 3</note>, <note>the center of the circle is the point C on the axis of rotation</note>. <note>The vertex of the cone is point A</note>, <note>the distance between vertex A and center C is the height of the cone, denoted as AC, which is 4</note>. <note>The lateral surface of the cone is a curved surface</note>, <note>the distance from vertex A to any point on the circumference is the slant height, denoted as AB, which is 5</note>.

Formula for the Surface Area of a Cone

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

In the diagram of this problem, the cone has a circular base with a <note>radius of 3</note> and a <note>base area of πx3²</note>. When the lateral surface of the cone is unfolded, it forms a sector with a <note>radius equal to the slant height of 5</note>, and the sector's <note>arc length is equal to the circumference of the base 2πx3</note>. The lateral surface area is equal to the area of the sector, which is <note>πx3×5</note>. The total surface area of the cone is the sum of the base area and the lateral surface area, so the total surface area is <note>πx3² + πx3×5</note>.

Generatrix

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

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


Question: Find the length of \widehat J K. Round to two decimal places.

Answer: 1.05

1. Given that the central angle ∠KCJ corresponding to arc length JK is 30°, using the conversion formula between degrees and radians, radian = degree * (π/180) = 30 * (π/180) = π/6.

2. Using the formula for the arc length of a sector to calculate the arc length JK, the arc length L of a sector is equal to the central angle θ (expressed in radians) multiplied by the radius r: L = θr. Substituting θ = π/6 and r = 2, we get L = π/3.

3. Taking the approximate value of π as 3.1415926, the calculated result rounded to two decimal places is finally 1.05.

4. Through the above reasoning, the final answer is 1.05 feet.

Definition of Circle

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

<note>In circle C, point C is the center</note>, with a radius of <note>2 feet</note>. All points in the figure that are <note>2 feet away from point C are on circle C</note>.

Definition of Central Angle

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

<note>The center of the circle is point C</note>, point J and point K are two points on the circle, <note>the angle ∠JCK formed by the lines CJ and CK is called the central angle</note>, its vertex is at <note>the center of the circle C</note>, and the angle is <note>30°</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 \( r \): \( L = \theta r \).

The length of the arc <note>JK</note> is the arc length corresponding to the calculated radians. The central angle <note>∠JCK is 30°</note>, and its radian measure is <note>θ (radians) = θ (degrees) × π / 180</note>, that is, <note>30° = 30 × π / 180 = π / 6</note>. Substituting the <note>radius of 2 feet</note> and the <note>radian measure of π / 6</note> into the arc length formula, the arc length = <note>2 × π / 6 = π / 3</note>.

Formula for Conversion between Degrees and Radians

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

<note>The angle of ∠KCJ is 30°</note>, according to the formula for conversion between degrees and radians, the conversion to radians = degrees * (π/180), that is <note>30 * (π/180)</note>.


Question: As shown in the figure, points B, E, C, F are on the same line, △ABC≌△DEF, ∠B=45°, ∠F=65°, then the degree measure of ∠COE is ()

Answer: 70°

1. Given △ABC≌△DEF, according to ##the definition of congruent triangles##, we get ∠BAC = ∠EDF, ∠ACB = ∠DFE, ∠ABC = ∠DEF.

2. Based on the given conditions ∠B = 45° and ∠F = 65°, we can obtain ∠DEF = ∠ABC = 45°, ∠ACB = ∠DFE = 65°.

3. ##Given that points B, E, C, and F are on the same line, and from the figure, point O is on the intersection line of AC and DE, then ∠OEC=∠DEF, ∠OCE=∠ACB##.

4. ##In △OEC##, according to the triangle angle sum theorem, ##we get ∠OEC+∠OCE+∠COE=180°, that is, ∠DEF+∠ACB+∠COE=180°, calculating we get## ∠COE = 180° - ∠DEF - ∠ACB.

5. Substituting the given angles into the formula, we calculate ∠COE = 180° - 45° - 65° = 70°.

6. Through the above reasoning, we finally get the answer as 70°.

普通三角形; 邻补角; 三角形的外角; 对顶角; 直线

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 ABC and triangle DEF are congruent triangles</note>, triangle ABC's <note>corresponding sides and corresponding angles are equal to those of triangle DEF</note>, namely:
<note>side AB = side DE</note>
<note>side BC = side EF</note>
<note>side AC = side DF</note>
At the same time, the corresponding angles are also equal:
<note>angle BAC = angle EDF</note>
<note>angle ACB = angle DFE</note>
<note>angle ABC = angle DEF</note>.

Triangle Angle Sum Theorem

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

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


Question: In the diagram above, lines EF and GH are parallel, and line AB is perpendicular to lines EF and GH. What is the length of line AB?

Answer: 5*\sqrt{3}

In the diagram above, lines EF and GH are parallel, and line AB is perpendicular to lines EF and GH. ∠DAF is 120°, and segment AC is 10.

According to the parallel axiom 2 and the definition of corresponding angles, ∠ACH is also 120°. Meanwhile, ∠ACB is the adjacent supplementary angle of ∠ACH, so ∠ACB = 180° - 120° = 60°.

According to the definition of perpendicular lines, since AB ⊥ GH, ∠ABC = 90°. According to the definition of a right triangle, in triangle ABC, ∠ABC = 90°, so triangle ABC is a right triangle.

According to the triangle angle sum theorem, ∠BAC = 180° - ∠ABC - ∠ACB = 180° - 90° - 60° = 30°.

According to the properties of a 30°-60°-90° triangle, AB = √3/2 AC = √3/2 * 10 = 5√3.

In conclusion, the final answer is option c, that is, the length of AB is 5√3.

平行线; 垂线; 直角三角形; 内错角

Definition of Parallel Lines

In the figure of this problem, <note>line EF and line GH are located in the same plane</note>, and <note>they have no intersection points</note>, therefore, according to the definition of parallel lines, <note>line EF and line GH are parallel lines</note>.

Parallel Postulate 2 of Parallel Lines

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

Original: <note>两条平行线EF和GH</note>被<note>第三条直线DC</note>所截，形成了以下几何关系：
1. 同位角：<note>角DAF和角ACH相等</note>。
2. 内错角：<note>角EAC和角ACH相等</note>。
3. 同旁内角：<note>角FAC和角ACH互补</note>，即<note>角FAC + 角ACH = 180度</note>。

这些关系说明了在<note>两条平行线被第三条直线所截时</note>，同位角相等，内错角相等，同旁内角互补。

Translation: <note>Two parallel lines EF and GH</note> are intersected by <note>a third line DC</note>, forming the following geometric relationships:
1. Corresponding angles: <note>Angle DAF and angle ACH are equal</note>.
2. Alternate interior angles: <note>Angle EAC and angle ACH are equal</note>.
3. Consecutive interior angles: <note>Angle FAC and angle ACH are supplementary</note>, that is, <note>Angle FAC + Angle ACH = 180 degrees</note>.

These relationships illustrate that when <note>two parallel lines are intersected by a third line</note>, corresponding angles are equal

Definition of Corresponding Angles

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

In the figure of this problem, <note>two parallel lines EF and GH</note> are intersected by <note>a line DC</note>, where <note>angle DAF and angle ACH are on the same side of the intersecting line CD and on the same side of the intersected lines EF and GH</note>, therefore <note>angle DAF and angle ACH are corresponding angles</note>. Corresponding angles are equal, that is, <note>angle DAF is equal to angle ACH</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 lines <note>AB and GH</note> intersect to form an <note>angle ∠ABC of 90 degrees</note>, so according to the <note>definition of perpendicular lines</note>, lines AB and GH are perpendicular to each other.

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 ACH and angle ACB share a common side AC</note>, <note>their other sides CH and CB are extensions in opposite directions</note>, so angles ACH and ACB are adjacent supplementary angles.

Definition of Right Triangle

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

Triangle Angle Sum Theorem

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

Properties of 30°-60°-90° Triangle

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

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


Question: In the figure, m \angle 3 = 110 and m \angle 12 = 55. Find the measure of \angle 13.

Answer: 55

1. Given ∠12 = 55°, the lines containing ∠10 and ∠12 are parallel.

2. Since the lines containing ∠10 and ∠12 are parallel, according to the definition of corresponding angles, ∠10 and ∠12 are corresponding angles; and according to the parallel lines axiom 2, ∠10 = ∠12 = 55°.

3. ∠13 and ∠10 are formed by the intersection of two lines, according to the definition of vertical angles, ∠10 = ∠13.

4. From the above information, ∠13 = ∠10 = ∠12 = 55°.

5. According to the calculations, the answer is 55°.

内错角; 平行线; 对顶角

Definition of Vertical Angles

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

Two intersecting lines form four angles: <note>∠9, ∠10, ∠13, and ∠14</note>. According to the <note>definition of vertical angles</note>, <note>∠13 and ∠10 are vertical angles</note>, and since vertical angles are equal, <note>∠13 = ∠10</note>.

Definition of Corresponding Angles

<note>∠10 and ∠12 are on two parallel lines cut by the same transversal</note>, <note>∠10 and ∠12 are on the same side of the transversal</note>, so <note>∠10 and ∠12 are corresponding angles</note>.

Parallel Postulate 2 of Parallel Lines

<note>∠10 and ∠12 are corresponding angles</note>, so <note>∠10 = ∠12</note>.


Question: As shown in the figure, given that the radius of the base of the cone is 6 and the slant height is 10, then the lateral surface area of this cone is ()

Answer: 60π

1. The radius of the base of the cone is 6, and the circumference of the base is twice the radius of the base multiplied by π, which is 2×6π.

2. Calculating, the circumference of the base is 12π.

4. The lateral surface area of the cone is equal to the circumference of the base multiplied by the slant height and then divided by 2, specifically calculated as: 1/2×12π×10.

5. Calculating, the lateral surface area of the cone is 60π.

Cone

<note>The base of the cone is a circle</note>, <note>the radius of the circle is 6</note>, <note>the center of the circle is the center point of the base</note>. <note>The vertex of the cone is the point at the top of the cone</note>, <note>the distance between the vertex and the center is the height of the cone</note>. <note>The lateral surface of the cone is a curved surface</note>, <note>the distance from the vertex to any point on the circumference is the slant height, which is 10</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.

In the diagram of this problem, <note>the development of the cone is a sector</note>. <note>The radius of the sector is the slant height of the cone, which is 10</note>, <note>the arc length of the sector is the circumference of the base circle of the cone, which is 12π</note>. Therefore, <note>the radius of the sector, 10, is equal to the slant height of the cone</note>, <note>the arc length of the sector, 12π, is equal to the circumference of the base circle of the cone, which is 2πr</note>, where <note>r is the radius of the base circle, which is 6</note>.

Circumference Formula of Circle

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

<note>The base of the cone is a circle</note>, <note>the radius of the circle is 6</note>. According to the circumference formula of the circle, <note>the circumference C of the circle is equal to 2π times the radius r, that is, C=2πr</note>, so the circumference of the base is 2π×6 = 12π.

Formula for the Surface Area of a Cone

Original: <note>In the figure of this problem, the base of the cone is a circle with a radius of 6</note>, <note>the base area is πx6?</note>. <note>The lateral surface of the cone, when unfolded, is a sector with a slant height of 10</note>, <note>the arc length of the sector is equal to the circumference of the base 2πx6</note>. <note>The lateral area is equal to the area of the sector, which is πx12×10</note>. The total surface area of the cone is equal to the base area plus the lateral area, so the total surface area is <note>πx6? + πx12×10</note>.


Question: As shown in the figure, line a ∥ b, point B is on line b, and AB ⊥ BC, ∠2 = 65°, then the degree of ∠1 is ()

Answer: 25°

1. From the given conditions, it is known that line a is parallel to line b.

2. According to the parallel lines axiom 2 and the definition of corresponding angles, ∠2 and ∠CBD are equal, so ∠CBD=∠2=65°.

3. According to the definition of perpendicular lines, AB⊥BC, ∠ABC=90°.

4. According to the definition of a straight angle, ∠1+∠ABC+∠CBD=180°.

5. Based on the given conditions, ∠ABC = 90°, ∠CBD=65°, so substituting in we get: ∠1+90°+65°=180°.

6. From this calculation we get: ∠1 = 180°-90° - 65° = 25°.

7. Therefore, the measure of angle 1 is 25°.

8. In summary, we finally obtain that the measure of ∠1 is 25°.

平行线; 内错角; 邻补角

Definition of Parallel Lines

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

<note>Line a and line b are located in the same plane</note>, and <note>they do not intersect</note>, so according to the definition of parallel lines, <note>line a and line b are parallel lines, denoted as a∥b</note>.

Definition of Straight Angle

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

<note>Ray BD</note> rotates around the endpoint <note>B</note> until it forms a straight line with the initial side, creating a straight angle. According to the <note>definition of straight angle</note>, the measure of a straight angle is <note>180 degrees</note>.

Definition of Perpendicular Lines

In the figure of this problem, <note>the angle ∠ABC formed by the intersection of line AB and line BC is 90 degrees</note>, therefore according to the definition of perpendicular lines, <note>line AB and line BC are perpendicular to each other</note>.

Parallel Postulate 2 of Parallel Lines

<note>Two parallel lines a and b are intersected by a third line BC</note>, forming the following geometric relationships: 1. <note>Corresponding angles: angle 2 and angle CBD are equal</note>. These relationships illustrate that <note>when two parallel lines are intersected by a third line</note>, corresponding angles are equal, alternate interior angles are equal, and same-side interior angles are supplementary.

Definition of Corresponding Angles

<note>In the diagram of this problem, two parallel lines a and b are intersected by a line BC, where angle 2 and angle CBD are on the same side of the intersecting line BC and on the same side of the intersected lines a and b, thus angle 2 and angle CBD are corresponding angles.</note> Corresponding angles are equal, so angle 2 is equal to angle CBD.


Question: As shown in the figure, AB∥CD, point E is on BC, and CD=CE, ∠D=74°, then the degree of ∠B is ()

Answer: 32°

1. Given AB∥CD, point E is on BC, and CD=CE.

2. Because CD=CE, in △CDE, ∠D and ∠E are corresponding angles. According to the properties of an isosceles triangle, the base angles are equal, thus, ##∠D = ∠CED= 74°##.

####

##3. According to the triangle angle sum theorem, the sum of the angles in △CDE is 180°. Therefore, we can write the equation: ∠D + ∠C + ∠CED = 180°##.

##4. Substitute the given conditions, replacing ∠D and ∠CED, we get: 74° + 74° + ∠C = 180°##.

####

##5. Solve the equation, we get ∠C = 32°##.

####

##6. According to the parallel line postulate 2, alternate interior angles are equal, ∠B=∠C= 32°##.

####

##7##. Through the above reasoning, the final answer is 53°.

平行线; 内错角; 普通三角形; 等腰三角形

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, in triangle CDE, side CD and side CE are equal</note>, therefore triangle CDE is an isosceles triangle.

Properties of Isosceles Triangle

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

<note>In the isosceles triangle CDE, sides CD and CE are equal</note>. Therefore, according to the properties of isosceles triangles, the angles opposite the equal sides are equal, i.e., <note>∠D = ∠DEC</note>.

Triangle Angle Sum Theorem

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

Parallel Postulate 2 of Parallel Lines

<note>Parallel lines AB and CD</note> are intersected by a third <note>line BC</note>, forming the following geometric relationship:
##1. <note>Alternate interior angles: ∠C and ∠B are equal</note>.


Question: As shown in the figure, there is an ancient rural pestle. It is known that the height of the support column AB is 0.3 meters, the length of the pedal DE is 1 meter, and the distance from the support point A to the stepping point D is 0.6 meters. Originally, the pestle head point E touched the ground, but now the stepping point D touches the ground. How much does the pestle head point E rise?

Answer: 0.5米

1. Given that the height of the support column AB is 0.3 meters, the length of the pedal DE is 1 meter, and the distance from the support point A to the pedal point D is 0.6 meters, according to the diagram, △DAB and △DEF can be drawn.

2. Given that angle ABD = angle EFD = 90 degrees, angle D is the common angle, therefore △DAB ∽ △DEF.

3. According to the properties of similar triangles, the ratios of corresponding sides are equal, so AD:DE = AB:EF.

4. Substituting the given data, we get 0.6:1 = 0.3:EF, thus EF can be calculated as 0.5 meters.

5. Since the original striking point E touches the ground, now the pedal point D touches the ground, so the height that the striking point E rises is equal to the length of EF, which is 0.5 meters.

普通三角形; 线段; 旋转

Definition of Similar Triangles

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

<note>Triangle DAB and triangle DEF are similar triangles</note>. According to the definition of similar triangles: <note>∠DAB = ∠DEF</note>, <note>∠ADB = ∠DFE</note>, <note>∠ABD = ∠EDF</note>; <note>AD/DE = AB/EF = DB/DF</note>.

Similarity Theorem for Triangles (AA)

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

In the diagram for this problem, in triangles ABD and EFD, if <note>angle D is equal to angle D</note>, and <note>angle ABD is equal to angle EFD</note>, then <note>triangle ABD is similar to triangle EFD</note>.


Question: As shown in the figure, place two vertices of a right triangle with a 45° angle on opposite edges of a straight ruler. If ∠1=27.5°, then ∠2 equals ()

Answer: 17.5°

1. Given AB is parallel to CD, let CF intersect AB at point G. According to the parallel axiom 2 and the definition of alternate interior angles, ∠AEC=∠1.

2. Given triangle FCE is a 45°-45°-90° right triangle, ∠FCE=∠AEC+∠2=45°.

3. Substituting the conclusion from step 1, ∠AEC=∠1=27.5°.

4. That is, 27.5°+∠2=45°, solving for ∠2=17.5°.

5. Through the above reasoning, the final answer is 17.5°.

直角三角形; 邻补角; 内错角

Definition of Parallel Lines

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

Definition of Right Triangle

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

<note>In the figure of this problem, in triangle FEC, angle EFC is a right angle (90 degrees)</note>, therefore triangle FEC is a right triangle. <note>Sides FE and FC are the legs, and side EC is the hypotenuse</note>.

Definition of Alternate Interior Angles

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

<note>Two parallel lines AB and CD</note> are intersected by a <note>line CE</note>, where <note>angle 1 and angle AEC are located between the two parallel lines and on opposite sides of the intersecting line CE</note>, hence <note>angle AEC and angle 1 are alternate interior angles</note>. Alternate interior angles are equal, that is, <note>angle AEC is equal to angle 1</note>.

Parallel Postulate 2 of Parallel Lines

<note>In the figure of this problem, two parallel lines AB and CD are intersected by a third line CE, forming the following geometric relationship:</note> 1. ##Alternate Interior Angles##: <note>Angle AEC is equal to angle 1.</note>####


Question: If W X Y Z is a kite, find W P.

Answer: \sqrt { 20 }

1. Given quadrilateral WXYZ is a kite, WX = 6, XP = 4. According to the properties of a kite, WY is perpendicular to XZ at point P, ∠XPW=90°.

2. According to the definition of a right triangle, triangle XPW is a right triangle.

3. According to the Pythagorean theorem, in triangle XPW, WX^2=XP^2+WP^2, then WP=√(WX^2-XP^2).

4. Substituting the known values, we get WP=√(WX^2-XP^2)=√(6^2-4^2).

6. According to the above calculation steps, WP = √20 = 2√5.

7. Through the above reasoning, the final answer is 2√5.

线段; 菱形; 垂直平分线

Properties of a Kite

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

<note>In the diagram of this problem, it is known that the diagonals XZ and WY of kite WXYZ intersect at point P</note>. According to the properties of a kite, its two diagonals <note>XZ and WY are perpendicular to each other, i.e., XZ⊥WY</note>, and one of the diagonals <note>WY is bisected by the other diagonal XZ, i.e., WP=YP</note>.

Definition of Right Triangle

In the figure of this problem, in triangle XPW, <note>angle XPW is a right angle (90 degrees)</note>, therefore <note>triangle XPW is a right triangle</note>. <note>Side WP and side XP are the legs</note>, <note>side WX is the hypotenuse</note>.

Pythagorean Theorem

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

<note>In the figure of this problem, in the right triangle WPX, ∠WPX is a right angle (90 degrees)</note>, <note>sides WP and XP are the legs</note>, <note>side WX is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>WX^2=XP^2+WP^2</note>, that is 6^2 = 4^2+WP^2, thus solving for the length of WX.


Question: Find the value of x in the figure.

Answer: 71

1. Let the polygon be ABCDEFG. According to the definition of exterior angles of a polygon, its exterior angles are: (x-20)°, 21°, 42°, 29°, (x+14)°, x°, (x-10)°.

2. According to the exterior angle sum theorem of polygons, the sum of exterior angles of any polygon is equal to 360°.

3. According to the exterior angle sum theorem of polygons, we can derive the equation: (x-20)° + 21° + 42° + 29° + (x+14)° + x° + (x-10)° = 360° => (x-20) + 21 + 42 + 29 + (x+14) + x + (x-10) = 360.

5. Perform addition and subtraction of constants on the right side to get: 4x + 76 = 360.

6. Move the constant term to the right side to get 4x = 360 - 76 = 284.

7. Solve the equation to get: x = 284 / 4.

9. Through the above reasoning, the final answer is 71.

普通多边形; 同旁内角; 内错角

Definition of Exterior Angle of a Polygon

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

In the problem diagram, one interior angle of polygon ABCDEFG is <note>∠GAB</note>. The angle formed by extending the adjacent sides <note>GA</note> and <note>AB</note> of this interior angle, <note>(x-20)°</note>, is called the exterior angle of the interior angle <note>GAB</note>. Similarly, for the interior angle <note>ABC</note>, the angle formed by extending the adjacent sides <note>AB</note> and <note>BC</note>, <note>21°</note>, is called the exterior angle of the interior angle <note>ABC</note>. By analogy, the exterior angles of polygon ABCDEFG are <note>(x-20)°</note>, <note>21°</note>, <note>42°</note>, <note>29°</note>, <note>(x+14)°</note>, <note>x°</note>, <note>(x-10)°</note> in sequence.

Exterior Angle Sum Theorem of Polygon

For any polygon, the sum of its exterior angles is equal to 360°.

In the figure of this problem, in the <note>polygon ABCDEFG</note>, the exterior angles at each vertex are <note>(x-20)°</note>, 21°, 42°, 29°, <note>(x+14)°</note>, <note>x°</note>, <note>(x-10)°</note>. According to the <note>Exterior Angle Sum Theorem of Polygon</note>, the sum of these exterior angles is equal to 360°, i.e., <note>(x-20)° + 21° + 42° + 29° + (x+14)° + x° + (x-10)° = 360°</note>.


Question: As shown in the figure, in ⊙O, the length of chord AB is 10, and the inscribed angle ∠ACB=45°, then the diameter AD of this circle is ()

Answer: 10√{2}

1. Given ∠ACB = 45°, according to the inscribed angle theorem, the central angle corresponding to the inscribed angle ∠ACB is 2 times ∠ACB. Therefore, ∠AOB = 2 * 45° = 90°.

2. In circle O, connect OA and OB. Since OA and OB are radii, ##so OA = OB. Also, since ∠AOB = 90°, according to the definition of an isosceles right triangle, triangle AOB is an isosceles right triangle##.

3. ##According to the Pythagorean theorem, the sides of an isosceles right triangle have the relationship OA^2 + OB^2 = 2OA^2 = AB^2. Rearranging gives √2OA = AB##.

4. ##Given AB = 10, so OA = 10/√2 = 5√2##.

5. ##Therefore, AD = 2OA = 10√2##.

6. ##Based on the above reasoning, the final answer is 10√2.##

圆; 弦; 圆周角; 直角三角形; 直线

Definition of Inscribed Angle

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

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

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

Definition of Central Angle

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

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

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.

The line segment AD passes through the center O of the circle <note>线段AD通过圆心O</note>, and the endpoints A and D intersect with the circle O <note>两端A和D与圆O相交</note>, therefore <note>AD是圆O的直径</note>.

Pythagorean Theorem

<note>Triangle AOB is a right triangle</note>, <note>∠AOB = 90°</note>, <note>OA and OB are the legs, AB is the hypotenuse</note>. According to the Pythagorean Theorem, <note>OA^2 + OB^2 = 2OA^2 = AB^2</note>.


Question: In order to measure the height of the school's flagpole AC, the math interest group of a certain school erected a 1.5-meter pole DF at point F, as shown in the figure. They measured the length of the shadow EF of DF to be 1 meter, and then measured the length of the shadow BC of the flagpole AC to be 6 meters. Therefore, the height of the flagpole AC is ()

Answer: 9米

1. Given that the height of the pole DF is 1.5 meters, the length of the shadow EF is 1 meter, and the length of the shadow BC of the flagpole AC is 6 meters.

2. Because the angle of incidence of the sunlight is the same, the pole and its shadow, as well as the flagpole and its shadow, form similar right triangles with their heights and shadow lengths.

3. According to the proportionality of corresponding sides in similar triangles: in △DEF and △ABC, ##AC/DF=BC/EF=AB/ED##.

4. From the proportionality theorem of corresponding sides in similar triangles, we can derive the proportional relationship: ##DF / AC = EF / BC##.

5. Substitute the known values into the above proportional relationship: ##1.5 /AC = 1 / 6##.

6. By solving the above equation, we can obtain AC = 1.5 × 6.

7. Finally, the calculated length of AC is 9 meters.

Definition of Similar Triangles

<note>Triangles DEF and ABC are similar triangles</note>. According to the definition of similar triangles: <note>∠DFE = ∠BCA</note>, <note>∠EDF = ∠CAB</note>, <note>∠DEF = ∠ABC</note>; <note>DF/AC = EF/BC</note>.

Similarity Theorem for Triangles (AA)

<note>Angle C is equal to angle F</note>, and <note>angle D is equal to angle A</note>, so <note>triangle ABC is similar to triangle DEF</note>.


Question: m \angle 4 = 2 y + 32 and m \angle 5 = 3 y - 3. Find y.

Answer: 35

1. Given ∠4 = 2y + 32 and ∠5 = 3y - 3. Let the vertical line on the left be l, the parallel line on the right be m, and the transversal line be n. The intersection point on the left is labeled as A, and the intersection point on the right is labeled as B.

2. According to the parallel postulate 2 of parallel lines, ∠4 and ∠8 are corresponding angles, so ∠4 = ∠8.

3. ∠5 and ∠8 are vertical angles, so ∠5 = ∠8.

4. Therefore, we have ∠4 = ∠8 = ∠5. Combining ∠4 = 2y + 32 and ∠5 = 3y - 3, we get 2y + 32 = 3y - 3.

5. Simplifying the equation, we get 35 = y.

6. Through the above reasoning, we finally obtain the answer y = 35.

Parallel Postulate 2 of Parallel Lines

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

<note>Two parallel lines l and m are intersected by a third line n</note>, forming the following geometric relationship: 1. Corresponding angles: <note>Angle 4 and Angle 8 are equal</note>.

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, two intersecting lines <note>m and n intersect at point B</note>, forming four angles: <note>angle 5, angle 6, angle 7, and angle 8</note>. According to the definition of vertical angles, <note>angle 5 and angle 8 are vertical angles</note>, <note>angle 6 and angle 7 are vertical angles</note>. Since vertical angles are equal, <note>angle 5 = angle 8</note>, <note>angle 6 = angle 7</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 l and m are intersected by a line n</note>, where <note>angle 4 and angle 8 are on the same side of the intersecting line n</note>, <note>on the same side of the two intersected lines l and m</note>, therefore angle 4 and angle 8 are corresponding angles. Corresponding angles are equal, that is, <note>angle 4 is equal to angle 8</note>.


Question: As shown in the figure, in Rt△ABC, it is known that ∠A=90°, AC=3, AB=4, then sinB equals ()

Answer: \frac{3}{5}

1. Given ∠A = 90°, AC = 3, AB = 4, ##according to the Pythagorean theorem, in Rt△ABC we have BC^2 = AC^2 + AB^2, i.e., BC = √(AC^2 + AB^2).##

2. Substituting the given conditions, ##we get BC = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.##

3. ##According to the definition of the sine function##, sinB = opposite side/hypotenuse, in Rt△ABC, sin∠ABC = AC/BC.

4. Substituting the given values, we get sin∠ABC = 3/5.

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

直角三角形; 正弦; 线段

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

Pythagorean Theorem

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


Question: As shown in the figure, AB is a ladder leaning against the wall. The distance between the foot of the ladder and the wall is 2 meters. The distance between point D on the ladder and the wall is 1.8 meters. The length of BD is 0.6 meters. Find the length of the ladder.

Answer: 6.00米

1. As shown in the figure, let the wall be vertical AC, the ground be horizontal BC, and AB be the ladder leaning against the wall. Point D is a point on the ladder AB such that DE is perpendicular to AC, and the distance between DE and the wall is 1.8 meters.

2. According to the problem statement, DE and BC are both perpendicular to AC, so DE is parallel to BC, i.e., DE∥BC.

3. Because DE∥BC, ##according to the parallel lines axiom 2, corresponding angles are equal, then ∠ADE=∠ABC##.

4. ##Also, as can be seen from the figure, ∠A is the common angle of triangles ADE and ABC, according to the similarity theorem (AA), we get## △ADE∽△ABC.

5. According to the definition of similar triangles, corresponding sides are proportional, i.e., ##AD:AB =DE:BC##.

6. According to the problem statement, BC=2 meters, DE=1.8 meters, ##BD=0.6 meters##. AB is the length of the ladder, let AD be x, ##then AB=x+0.6##.

7. ##Substitute the known data into the proportion equation to get## x:x+0.6 = 1.8:2.

8. ##Solve the proportion equation to get: x=5.4##.

10. ##Obviously, the length of the ladder AB=6 meters##.

11. ##Finally, confirm that AB = 6 meters completely meets the problem statement##.

Similarity Theorem for Triangles (AA)

In the figure of this problem, in triangles ADE and ABC, if <note>angle ADE is equal to angle ABC</note>, and <note>angle DAE is equal to angle BAC</note>, then <note>triangle ADE is similar to triangle ABC</note>.

Definition of Perpendicular Lines

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

<note>DE and AC intersect to form the angle ∠DEA which is 90 degrees</note>, <note>BC and AC intersect to form the angle ∠BCA which is 90 degrees</note>, therefore according to the definition of perpendicular lines, <note>DE and AC are perpendicular to each other</note>, <note>BC and AC are perpendicular to each other</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 DE and line BC are located in the same plane</note>, <note>and they do not intersect</note>, therefore according to the definition of parallel lines, <note>line DE and line BC are parallel lines</note>.

Definition of Similar Triangles

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

<note>Triangle ADE and triangle ABC are similar triangles</note>. According to the definition of similar triangles: <note>∠ADE = ∠ABC</note>, <note>∠DEA = ∠BCA</note>, <note>∠EAD = ∠CAB</note>; <note>AB/AD = BC/DE</note>.


Question: As shown in the figure, in a badminton match, the player Lin Dan standing at point M hits the shuttlecock from point N to point B on the opponent's side. Given that the net height OA = 1.52m, OB = 4m, OM = 5m, find the distance from the hitting point to the ground NM = ()

Answer: 3.42m

1. Given AO is perpendicular to BM, NM is perpendicular to BM, ##according to the definition of perpendicular lines, ∠AOB=NMB=90°##.

2. ##According to the theorem of similar triangles (AA), since ∠AOB=NMB, ∠ABO=NBM, we can conclude that △ABO is similar to △NBM. Based on the definition of similar triangles##, we can obtain AO/NM = OB/BM.

3. According to the given conditions, OA = 1.52 meters, OB = 4 meters, OM = 5 meters.

4. Calculate BM, BM = OB + OM = 4 meters + 5 meters = 9 meters.

5. Based on the proportional relationship AO/NM = OB/BM, we get 1.52/NM = 4/9.

6. Solve this proportion equation to get NM = 3.42 meters.

7. Through the above reasoning, we finally conclude that NM is 3.42 meters.

直角三角形; 点; 线段

Definition of Perpendicular Lines

In the figure of this problem, <note>line segment AO is perpendicular to line segment BM</note>, <note>line segment NM is perpendicular to line segment BM</note>, corresponding to the definition of two perpendicular line segments, forming <note>two 90-degree angles, ∠AOB and ∠NMB</note>.

Definition of Similar Triangles

<note>In the diagram of this problem, triangle ABO and triangle NBM are similar triangles</note>. According to the definition of similar triangles: <note>∠BAO = ∠BNM</note>, <note>∠ABO = ∠NBM</note>, <note>∠AOB = ∠NMB</note>; <note>AO/NM = OB/BM</note>.

Similarity Theorem for Triangles (AA)

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

In the figure of this problem, triangles <note>AOB</note> and <note>NBM</note>, if <note>angle AOB is equal to angle AMB</note>, and <note>angle NBM is equal to angle ABO</note>, therefore <note>triangle ABO is similar to triangle NBM</note>.


Question: Tangent MP is drawn to \odot O. If MO = 20, find x.

Answer: 12

1. Given that line segment MP is a tangent to the circle with center O, according to the property of tangents, the tangent is perpendicular to the radius passing through the point of tangency, i.e., ##∠MPO## is a right angle.

2. Since MO = 20 and MP = 16, using the Pythagorean theorem in right triangle MOP, i.e., MO^2 = MP^2 + OP^2, we can find that OP is the radius of the circle, and OP = x.

3. Substitute the given values into the Pythagorean theorem formula, yielding 20^2 = 16^2 + x^2.

4. Calculate the specific values: 400 = 256 + x^2.

5. Solve this equation to get x^2 = 400 - 256 = 144.

6. Take the positive value of x, obtaining x = ##√144## = 12.

7. The above calculations show that the final answer is 12.

切线; 圆; 直角三角形

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 MP have only one common point P</note>, this common point is called the point of tangency. Therefore, <note>line MP is the tangent to circle O</note>.

Definition of Right Triangle

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

<note>In the diagram of this problem, in triangle MOP, angle MPO is a right angle (90 degrees)</note>, therefore triangle MOP is a right triangle. <note>Side MP and side OP are the legs, side MO is the hypotenuse</note>.

Property of the Tangent Line to a Circle

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

In the figure of this problem, in circle O, point P is the point of tangency of the line <note>MP</note> with the circle, and the segment <note>OP</note> is the radius of the circle. According to the property of the tangent line to a circle, the tangent line <note>MP</note> is perpendicular to the radius <note>OP</note> at the point of tangency <note>P</note>, that is, <note>∠MPO = 90°</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 <note>right triangle MOP</note>, <note>∠MPO is a right angle (90 degrees)</note>, sides <note>MP and OP are the legs</note>, side <note>MO is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>MO^2 = MP^2 + OP^2</note>.


Question: Find h in the triangle.

Answer: 11

1. Let the triangle be ABC. According to the definition of a right triangle, △ABC is a right triangle with the leg AB = h and the hypotenuse AC = 22. According to the definition of the sine function, sin(30°) = opposite side AB/hypotenuse AC, we get sin(30°) = h/22.

2. Calculate sin(30°) = 1/2 and substitute it into the above equation to get 1/2 = h/22.

3. By multiplying both sides of the equation by 22, we get h = 22 * (1/2).

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

Definition of Right Triangle

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

Definition of Sine Function

In the right triangle <note>ABC</note>, <note>the 30° angle is an acute angle</note>, <note>side AB is the opposite side of the 30° angle</note>, <note>side AC is the hypotenuse</note>. According to the definition of the sine function, <note>the sine value of the 30° angle is equal to the ratio of the opposite side AB to the hypotenuse AC</note>. From the diagram, we know <note>AB=h, AC=22</note>, thus <note>sin(30°) = AB/AC=h/22</note>.


Question: As shown in the figure, in Rt△ABC, ∠C=90°, sinA=\frac{1}{3}, then the value of \cosB is ()

Answer: \frac{1}{3}

1. Given ∠ACB=90°, this is a right triangle.

2. Since the sum of the interior angles of a triangle is 180°, according to the triangle sum theorem, we get: ∠BAC + ∠ABC + ∠ACB = 180°.

3. Substituting the given condition ∠ACB = 90°, we get ∠BAC + ∠ABC = 90°.

4. Because it is given that sin(∠BAC) = \frac{1}{3}, we know the sine and cosine relationship between complementary angles, i.e., sin(∠BAC) = cos(∠ABC). This is derived from the properties of trigonometric functions of complementary angles.

5. Therefore, we get cos(∠ABC) = \frac{1}{3}.

6. Through the above reasoning, we finally get the answer as \frac{1}{3}.

直角三角形; 正弦; 余弦

Definition of Right Triangle

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

Definition of Sine Function

In the <note>right triangle ABC</note>, <note>angle BAC is an acute angle</note>, <note>side BC is the opposite side of angle BAC</note>, and <note>side AB is the hypotenuse</note>. According to the <note>definition of sine function</note>, the <note>sine value of angle BAC</note> is equal to the <note>ratio of the opposite side BC to the hypotenuse AB</note>, that is, <note>sin(∠BAC) = BC / AB = 1 / 3</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 diagram of this problem, in the right triangle ABC, <note>side BC is the adjacent side of angle ∠ABC</note>, <note>side AB is the hypotenuse</note>. Therefore, according to the definition of the cosine function, <note>the cosine value of angle ∠ABC is equal to the ratio of the adjacent side BC to the hypotenuse AB, i.e., cos(∠ABC) = BC / AB</note>.

Triangle Angle Sum Theorem

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

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


Question: In Rt△ABC, ∠ACB=90°, CD⊥AB at point D, if AC=3, BC=4, then tanα equals ()

Answer: \frac{3}{4}

1. In triangles ABC and ACD, ∠A=∠A, ∠ACB=∠ADC=90°, so triangle ABC~triangle ACD (similar triangles criterion (AA)), thus ∠α=∠B

2. According to the definition, tanα represents the ratio of the length of the opposite side to the length of the adjacent side. In this figure, tanα=tan∠B.

3. In right triangle ABC, tan∠B=length of the opposite side/length of the adjacent side=AC/BC. Given AC=3 and BC=4, we can find tan∠B=3/4.

4. Through the above reasoning, we finally obtain the value of tanα as 3/4.

直角三角形; 垂线; 正切

Definition of Right Triangle

In the diagram of this problem, in triangle ABC, <note>angle ACB is a right angle (90 degrees)</note>, therefore <note>triangle ABC is a right triangle</note>. <note>Side AC and side BC are the legs</note>, <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 <note>right triangle ABC</note>, <note>angle ∠ABC is an acute angle</note>, <note>side AC is the opposite side of angle ∠ABC</note>, <note>side BC is the adjacent side of angle ∠ABC</note>, so the <note>tangent value of angle ∠ABC</note> is equal to the <note>length of side AC divided by the length of side BC</note>, that is, <note>tan(∠ABC) = AC / BC</note>.

Similarity Theorem for Triangles (AA)

In the given diagram, triangle ABC and triangle ACD, angle A is equal to angle A, and angle ACB is equal to angle ADC, so triangle ABC <note>is similar to triangle ACD</note>.

Definition of Perpendicular Lines

Original text: <note>The angle ∠ADC formed by the intersection of line CD and line AB is 90 degrees</note>, therefore according to the definition of perpendicular lines, <note>line CD and line AB are perpendicular to each other</note>.

Definition of Similar Triangles

In this problem's diagram, <note>triangle ACD and triangle ABC are similar triangles</note>. According to the definition of similar triangles: <note>angle ACD = angle B</note>, <note>angle A = angle A</note>, <note>angle ACB = angle ADC</note>; <note>AC/AB = AD/AC = CD/BC</note>.


Question: Solve for x.

Answer: 5

1. The given conditions are ##angle HEG = 8x + 6 and angle HGF = 10x - 4##.

2. According to ##Corollary 1 of the Inscribed Angle Theorem, in a circle, any two inscribed angles that correspond to the same arc are equal##.

3. Because ##angle HEF and angle HFG correspond to arc HG##.

4. Therefore, ##angle HEF and angle HFG## are equal, thus we get the equation: 8x + 6 = 10x - 4.

5. Solving this equation: First, rearrange the terms: 8x + 6 = 10x - 4, which simplifies to 10x - 8x = 6 + 4.

7. Next, solving this equation, we get x = 5.

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

圆; 圆周角; 圆内接四边形

Definition of Inscribed Angle

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

<note>In the diagram of this problem, in circle EFGH, the vertex E of angle EHF is on the circumference, and the two sides of angle EHF intersect circle EFGH at points H and F</note>. Therefore, angle EHF is an inscribed angle. Similarly, <note>the vertex F of angle FGE is on the circumference, and the two sides of angle FGE intersect circle EFGH at points G and E</note>. Therefore, angle FGE is also an inscribed angle. <note>In the diagram of this problem, in circle EFGH, the vertex (point E) of angle HEG is on the circumference, and the two sides of angle HEG intersect circle EFGH at points H and G</note>. Therefore, angle HEG is an inscribed angle. <note>In the diagram of this problem, in circle EFGH, the vertex (point H) of angle HFG is on the circumference, and the two sides of angle HFG intersect circle EFGH at points H and G</note>. Therefore, angle HFG is an inscribed angle.

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

<note>In the figure of this problem, in circle EFGH, the inscribed angles ∠HEG and ∠HFG corresponding to arc HG are equal</note>. According to Corollary 1 of the Inscribed Angle Theorem, <note>the inscribed angles ∠HEG and ∠HFG corresponding to the same arc HG are equal</note>, that is, <note>∠HEG = ∠HFG</note>.


Question: As shown in the figure, AB is the diameter of circle O, CD is the chord of circle O, and the extensions of AB and CD intersect at point E. Given AB = 2DE and ∠E = 16°, find the degree measure of ∠ABC.

Answer: 24°

1. Connect OD. Given that AB is the diameter of circle O, according to the definition of diameter, we get AB=2OD.

2. From the given condition AB=2DE, combined with the above conclusion AB=2OD, we can deduce that OD=DE.

3. In triangle DOE, since OD=DE, according to the definition of an isosceles triangle, triangle DOE is an isosceles triangle.

4. According to the properties of an isosceles triangle, ∠DOE=∠DEO.

5. ∠DOE is the central angle corresponding to arc BD, ∠DCB is the inscribed angle corresponding to arc BD. According to the inscribed angle theorem, ∠DCB=∠DOE/2=∠DEO/2=8°.

6. ∠ABC is the exterior angle of triangle CBE. According to the exterior angle theorem of triangles, ∠ABC=∠DCB+∠E=24°.

7. Through the above reasoning, the final answer is 24°.

圆; 圆周角; 弦; 直线; 等腰三角形

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 of circle O</note>, connecting the center of the circle <note>O</note> and the two points <note>A, B</note> on the circumference, with a length equal to twice the radius, that is <note>AB = 2 * OA</note>.

Definition of Chord

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

In the figure of this problem, in circle O, <note>point C and point 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>.

Inscribed Angle Theorem

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

<note>∠DCB is the inscribed angle subtended by chord BD</note>. According to the Inscribed Angle Theorem, the inscribed angle is equal to half of its corresponding central angle, which means <note>∠DCB = 1/2∠DOB = 1/2 × 16° = 8°</note>.

Exterior Angle Theorem of Triangle

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

<note>In the figure of this problem, in triangle BCE, angle ∠ABC is an exterior angle of the triangle</note>, angles ∠BCE and ∠BEC are the two non-adjacent interior angles to the exterior angle ∠ABC. According to the Exterior Angle Theorem of Triangle, <note>the exterior angle ∠ABC is equal to the sum of the two non-adjacent interior angles ∠BCE and ∠BEC, that is, angle ∠ABC = angle ∠BCE + angle ∠BEC.</note>

Definition of Isosceles Triangle

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

In the figure of this problem, in triangle ODE, <note>side DO and side DE are equal</note>, therefore <note>triangle ODE is an isosceles triangle</note>.

Properties of Isosceles Triangle

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

<note>In the isosceles triangle ODE, sides DO and DE are equal</note>. Therefore, according to the properties of an isosceles triangle, <note>the angles opposite the equal sides are equal</note>, that is, <note>angle DOE = angle DEO</note>.

Definition of Central Angle

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

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

Definition of Inscribed Angle

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

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

Original text: <note>In the figure of this problem, an interior angle of polygon BCE is ∠CBE</note>, extending the adjacent sides of this interior angle <note>EB and CB forming the angle ∠ABC is called the exterior angle of the interior angle ∠BCE</note>.

Properties of Isosceles Triangle

<note>In the figure of this problem, in the isosceles triangle ODE, side OD and side DE are equal</note>. Therefore, according to the properties of an isosceles triangle, <note>the angles opposite the equal sides are equal</note>, that is, <note>angle DOE = angle DEO</note>.


Question: Find the degree measure of \angle Z, accurate to one decimal place.

Answer: 33.7

1. Given that YX = 5, XZ = 9, ∠Y=90°, according to the definition of a right triangle, △XYZ is a right triangle, YX is the leg, and XZ is the hypotenuse.

2. According to the definition of the sine function, sin(∠XZY) = YX / XZ = 5 / 9.

3. Calculate ∠XZY using the arcsine function: ∠XZY = arcsin(5/9).

4. Use a calculator to solve: ∠XZY ≈ 33.7 degrees (rounded to the nearest tenth).

5. Through the above reasoning, the final answer is 33.7 degrees.

直角三角形; 正弦; 余弦; 正切

Definition of Right Triangle

In the figure of this problem, in triangle XYZ, <note>angle Y is a right angle (90 degrees)</note>, therefore triangle XYZ is a right triangle. <note>Side YX and side YZ are the legs</note>, <note>side XZ is the hypotenuse</note>.

Definition of Sine Function

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

<note>∠XZY is an acute angle</note>, <note>side YX is the opposite side of ∠XZY</note>, <note>side XZ is the hypotenuse</note>. According to the definition of the sine function, <note>the sine value of ∠XZY is equal to the ratio of the opposite side YX to the hypotenuse XZ</note>, that is, <note>sin(∠XZY) = YX / XZ</note>.


Question: In ⊙O, AB is the diameter, CD is a chord, ∠ABD=28°, then the degree of ∠C is ()

Answer: 62°

2. According to the problem statement, AB is the diameter, ##∠ADB is the inscribed angle subtended by the diameter, therefore according to (Corollary 2 of the Inscribed Angle Theorem) the inscribed angle subtended by the diameter is a right angle##, ∠ADB=90°.

3. Given ∠ABD=28°, then by the triangle angle sum theorem, ∠DAB=##180°-##90°-28°=62°.

4. ##∠C and ∠DAB are both inscribed angles subtended by arc DB, according to Corollary 1 of the Inscribed Angle Theorem, ∠C=∠DAB##.

5. In conclusion, the measure of ∠C is 62°.

圆周角; 圆; 弦; 线段

Definition of Inscribed Angle

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

Definition of Diameter

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

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

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

<note>In the figure of this problem, in circle O, the angle subtended by the diameter AB at the circumference, ∠ADB, is a right angle (90 degrees)</note>.

Triangle Angle Sum Theorem

In the figure of this problem, in triangle ABD, <note>∠ABD</note>, <note>∠ADB</note>, and <note>∠DAB</note> are the three interior angles of triangle ABD. According to the Triangle Angle Sum Theorem, <note>∠ABD + ∠ADB + ∠DAB = 180°</note>.

In the figure of this problem, in circle O, the inscribed angles <note>∠DAB</note> and <note>∠DCB</note> corresponding to the arc <note>DB</note> are equal. According to Corollary 1 of the Inscribed Angle Theorem, the inscribed angles <note>∠DAB</note> and <note>∠DCB</note> corresponding to the same arc <note>DB</note> are equal, i.e., <note>∠DAB = ∠DCB</note>.


Question: As shown in the figure, AB is the diameter of ⊙O, chord CD ⊥ AB at E, connect OC and AD, and ∠A = 35°, then ∠AOC = ()

Answer: 110°

1. Connect OD, since ∠A = 35°, according to the inscribed angle theorem, the inscribed angle is half of the central angle that subtends the same arc, we get ∠BOD = 2∠A = 70°.

2. Given that AB is the diameter of ⊙O, chord CD is perpendicular to AB at E, according to the right triangle congruence criterion (hypotenuse, leg), we know triangle ODE is congruent to triangle OCE.

3. According to the definition of congruent triangles, ∠BOC = ∠BOD = 70°.

4. According to the definition of supplementary angles, ∠AOC + ∠BOC = 180°.

5. Based on the previous reasoning, we get ∠BOC = ∠BOD = 70°.

6. Substitute into the formula from step 4: ∠AOC = 180° - ∠BOC = 110°.

7. Through the above reasoning, the final answer is 110°.

圆; 圆心角; 直角三角形; 垂线; 弦

Definition of Diameter

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

Definition of Chord

In the diagram of this problem, in circle O, <note>points 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>.

Definition of Central Angle

In the figure of this problem, <note>∠AOC and ∠BOD are central angles</note>. Their vertices are both at <note>O</note>, and their sides are <note>OA, OC and OB, OD</note> respectively.

Definition of Perpendicular Lines

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

<note>Line CD and line AB intersect to form an angle ∠CEA of 90 degrees</note>, therefore according to the <note>Definition of Perpendicular Lines</note>, <note>line CD and line AB are perpendicular to each other</note>.

Inscribed Angle Theorem

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

Definition of Inscribed Angle

In the figure of this problem, in circle O, <note>the vertex of angle BAD (point A) 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>.

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

In the <note>right triangles ODE and OCE</note>, assume that <note>angles ODE and OEC are both right angles (90 degrees)</note>. If <note>side OD is equal to side OC (hypotenuse)</note>, and <note>side OE is equal to side OE (one leg)</note>, then according to the <note>Right Triangle Congruence (Hypotenuse, One Leg)</note>, these two <note>right triangles ODE and OCE 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 ODE and OCE are congruent triangles</note>, the corresponding sides and corresponding angles of triangle ODE are equal to those of triangle OCE, that is: <note>angle DOE = angle COE</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.

The original text: <note>Angle BOC and angle AOC share a common side OC</note>, and their <note>other sides OA and OB are extensions of each other in opposite directions</note>, so <note>angle BOC and angle AOC are adjacent supplementary angles</note>.


Question: As shown in the figure, points A, B, and C are on ⊙O. ∠ABO=40°, ∠ACO=30°, then the degree of ∠BOC is ()

Answer: 140°

1. According to the problem, draw the diameter of ⊙O through point A, intersecting ⊙O at point D.

2. In △OAB, since OA = OB, △OAB is an isosceles triangle, which means ∠OBA = ∠OAB.

3. Given that ∠ABO = 40°, therefore ∠OBA = ∠OAB = 40°.

4. In △OAB, ∠BOD is an exterior angle, according to the exterior angle theorem, ∠BOD = ∠ABO + ∠OAB = 2 × 40° = 80°.

5. Similarly, in △OAC, since OA = OC, △OAC is an isosceles triangle, similarly we have ∠OCA = ∠OAC.

6. Given that ∠ACO = 30°, therefore ∠OCA = ∠OAC = 30°.

7. In △OAC, ∠COD is an exterior angle, according to the exterior angle theorem, ∠COD = ∠ACO + ∠OAC = 2 × 30° = 60°.

8. From the previous reasoning, ∠BOC = ∠BOD + ∠COD = 80° + 60° = 140°.

9. After the above reasoning, the final answer is 140°.

Definition of Isosceles Triangle

<note>In the figure of this problem, in triangle OAB, side OA and side OB are equal</note>, therefore triangle OAB is an isosceles triangle. Similarly, <note>in triangle OAC, side OA and side OC are equal</note>, therefore triangle OAC is also an isosceles triangle.

Definition of Circle

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

<note>In circle O, point O is the center of the circle</note>, the radius is <note>OA</note>. All points in the figure that are at a distance equal to OA from <note>point O are on circle O</note>, that is, <note>A, B, C are all on the circumference</note>, and <note>OA = OB = OC</note>.

In the diagram of this problem, <note>an interior angle of polygon OAB is ∠AOB</note>, the angle <note>∠BOD</note> formed by extending the adjacent sides <note>AO</note> and <note>OB</note> of this interior angle is called the exterior angle of the interior angle <note>∠AOB</note>. Similarly, <note>an interior angle of polygon OAC is ∠AOC</note>, the angle <note>∠COD</note> formed by extending the adjacent sides <note>AO</note> and <note>OC</note> of this interior angle is called the exterior angle of the interior angle <note>∠AOC</note>.

Properties of Isosceles Triangle

<note>In the isosceles triangle OAB, sides OA and OB are equal</note>. Therefore, according to the properties of isosceles triangles, the angles opposite the equal sides are equal, that is, <note>∠OAB = ∠OBA = 40°</note>. Similarly, <note>in the isosceles triangle OAC, sides OA and OC are equal</note>. Therefore, according to the properties of isosceles triangles, the angles opposite the equal sides are equal, that is, <note>∠OAC = ∠OCA = 30°</note>.

Exterior Angle Theorem of Triangle

<note>In triangle OAB, ∠BOD is an exterior angle of triangle OAB</note>, <note>∠OAB and ∠OBA are two non-adjacent interior angles</note>, according to the exterior angle theorem of triangles, <note>the exterior angle ∠BOD is equal to the sum of the two non-adjacent interior angles ∠OAB and ∠OBA</note>, that is, <note>∠BOD = ∠OAB + ∠OBA = 40° + 40° = 80°</note>; similarly in <note>triangle OAC, ∠COD is an exterior angle of triangle OAC</note>, <note>∠OAC and ∠OCA are two non-adjacent interior angles</note>, according to the exterior angle theorem of triangles, <note>the exterior angle ∠COD is equal to the sum of the two non-adjacent interior angles ∠OAC and ∠OCA</note>, that is, <note>∠COD = ∠OAC + ∠OCA = 30° + 30° = 60°</note>.


Question: What is the value of x?

Answer: 60

1. Let the vertex of the X° angle be A, the vertex of the 70° angle be B, and the vertex of the 130° exterior angle be C.

2. Given ∠ABC = 70°, the exterior angle of triangle ACB = 130°.

3. According to the exterior angle theorem of triangles, any exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. Applied to this problem, the exterior angle of triangle ACB is the sum of ∠CAB = x° and ∠ABC = 70°.

4. Based on the exterior angle theorem of triangles, we can derive the equation: 130° = x° + 70°, thus concluding x° = 130° - 70°.

5. Through the above reasoning, the final answer is 60.

三角形的外角; 普通三角形

In the figure of this problem, <note>one interior angle of triangle ABC is ∠ACB</note>, extending the adjacent sides of this interior angle <note>AC</note> and <note>BC</note> forms an angle called the exterior angle of the interior angle ∠ACB, which is <note>130° angle</note>.

In triangle ABC, the 130° angle is an exterior angle of triangle ABC. ∠CAB = x° and ∠ABC = 70° are the two non-adjacent interior angles to the exterior angle 130°. According to the Exterior Angle Theorem of Triangle, the exterior angle 130° is equal to the sum of the two non-adjacent interior angles ∠CAB and ∠ABC, i.e., 130° = ∠CAB + ∠ABC.


Question: As shown in the figure, points A, B, and C are all on ⊙O. ∠ABO=50°, then ∠ACB=()

Answer: 40°

1. Given that points A, B, and C are on ⊙O, ∠ABO=50°.

2. Based on the definition of radius, it is known that OA and OB are radii of circle O, and OA=OB.

3. Since OA=OB, according to the definition of an isosceles triangle, triangle AOB is an isosceles triangle.

4. According to the properties of an isosceles triangle, we get ∠OAB = ∠ABO = 50°.

5. Based on the triangle angle sum theorem, ∠ABO + ∠OAB + ∠AOB = 180°, i.e., 50° + 50° + ∠AOB = 180°, we get ∠AOB = 180° - 50° - 50° = 80°.

6. According to the inscribed angle theorem, in the same circle, the measure of the central angle corresponding to the same arc is twice the measure of the inscribed angle. Therefore, ∠ACB = ∠AOB/2.

7. Substituting the values, we get ∠ACB = 80°/2 = 40°.

8. Through the above reasoning, we finally get ∠ACB = 40°.

Definition of Radius

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

In the diagram of this problem, in circle ⊙O, point O is the center of the circle, and points A and B are any points on the circle, line segments OA and OB are line segments from the center of the circle to any point on the circle, therefore line segments OA and OB are the radius of the circle.

Definition of Isosceles Triangle

In triangle OAB, side OA and side OB are equal because they are the radii of circle ⊙O, therefore triangle OAB is an isosceles triangle.

Definition of Inscribed Angle

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

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

Triangle Angle Sum Theorem

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

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

Inscribed Angle Theorem

<note>In circle O</note>, points A, B, and C are on the circle, <note>the central angle corresponding to arc AB and arc AC is ∠AOB</note>, <note>the inscribed angle is ∠ACB</note>. According to the Inscribed Angle Theorem, <note>∠ACB is equal to half of the central angle ∠AOB corresponding to arc AB</note>, i.e., <note>∠ACB = 1/2 ∠AOB</note>.

Properties of Isosceles Triangle

In the figure of this problem, <note>in the isosceles triangle AOB, sides OA and OB are equal</note>. Therefore, according to the properties of isosceles triangles, <note>the angles opposite the equal sides are equal, that is, angle ABO = angle OAB</note>.

Definition of Central Angle

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


Question: In parallelogram M N P Q, m \angle M = 10 x and m \angle N = 20 x, find \angle M.

Answer: 60

1. Given the parallelogram MNPQ, ##∠M = 10x## and ##∠N = 20x##. According to the theorem of supplementary adjacent angles in a parallelogram, adjacent angles are supplementary, i.e., ##∠M + ∠N = 180°##.

2. From the conclusion in step 1, ##∠M + ∠N = 180°##, substituting the given ##∠M = 10x## and ##∠N = 20x##, we get 10x + 20x = 180°.

3. Solving the equation 30x = 180°, we get x = 6.

4. Substituting ##∠M = 10x##, we get ##∠M## = 10 * 6 = 60°.

5. Through the above reasoning, the final answer is 60°.

Definition of Parallelogram

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

<note>Side MQ</note> is parallel to and equal to <note>Side NP</note>.

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

In <note>parallelogram MNPQ</note>, <note>angle M and angle N are adjacent interior angles</note>, <note>angle N and angle P are also adjacent interior angles</note>. According to the Adjacent Angles Supplementary Theorem of Parallelogram, <note>angle M + angle N = 180°</note>, <note>angle N + angle P = 180°</note>.


Question: In the triangle in the figure above, what is the value of x?

Answer: 2*\sqrt{3}

1. Let the vertex of the 60° angle in the triangle be point A, the vertex of the right angle be point B, and the other vertex of the triangle be point C, so ∠A = 60°, ∠B = 90°. According to the triangle angle sum theorem, ∠C = 180° - 90° - 60° = 30°.

2. Given that one leg of the right triangle BC = 6, the length of the other leg AB can be found.

3. In a 30°-60°-90° right triangle, according to the properties of a 30°-60°-90° triangle, the lengths of sides AB, BC, and AC are in the ratio 1:√3:2.

4. In this problem, BC is the √3 times corresponding side, so the side AB adjacent to the 60° angle should be 1 times, i.e., x = 6/√3.

5. After simplification, x = 6*(√3 / 3) = 2√3. This conclusion is derived from the side length ratio theorem of a 30°-60°-90° triangle.

6. Therefore, according to the ratio theorem, x = 2√3.

Definition of Right Triangle

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

In the diagram of this problem, there is a right angle (<note>angle B</note>) in the triangle (90 degrees), so the triangle is a right triangle. <note>BC and AB are the legs, and AC is the hypotenuse.</note>

Properties of 30°-60°-90° Triangle

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

In the figure of this problem, <note>in the 30°-60°-90° triangle ACB, side AC is the hypotenuse, side AB is opposite the 30-degree angle, and side BC is opposite the 60-degree angle</note>. According to the properties of a 30°-60°-90° triangle, <note>side AB is equal to half of side AC</note>, <note>side BC is equal to √3 times side AB</note>. That is: <note>AB = 1/2 * AC</note>, <note>BC = √3 * AB</note>.

Triangle Angle Sum Theorem

In the figure of this problem, in the triangle, <note>∠A=60°</note>, <note>∠B=90°</note>, according to the Triangle Angle Sum Theorem, <note>∠C=180°-∠B-∠A=180°-90°-60°=30°</note>.


Question: Find \angle C of quadrilateral ABCD.

Answer: 90

1. According to the right angle symbol in the figure, we know BC is perpendicular to CD, therefore \angle C = 90 degrees.

2. The final conclusion is: \angle C is 90.

普通四边形; 邻补角; 直角三角形

Definition of Perpendicular Lines

<note>Line BC and line DC intersect to form a 90-degree angle ∠C</note>, therefore, according to the definition of perpendicular lines, <note>line BC and line DC are perpendicular to each other</note>.


Question: AB \cong DF. Solve for x.

Answer: 2

1. Given that segment AB ##≌## segment DF, that is, segment AB and segment DF are congruent, which means their lengths are equal.

2. ##Let the center of the circle in the figure be O, the left extension of line CE intersects the circle O at point M, and the right extension of line CE intersects the circle O at point N. From the markings in the figure, it is known that segment CE is perpendicular to chord AB and chord DF. Since segment CE coincides with the diameter of circle O, according to the perpendicular bisector theorem, if a diameter is perpendicular to a chord in circle O, then the diameter bisects that chord. Therefore, the diameter bisects the chords AB and DF in circle O, that is, segment CE is the perpendicular bisector of chords AB and DF. And the diameter intersects chords AB and DF at points C and E, so AC = BC = 1/2 AB, DE = FE = 1/2 DF. Also, since segment AB and segment DF are congruent, that is, AB = DF, so BC = DE##.

3. ##Given BC = 9x and DE = 2x + 14, substituting BC = DE yields: 9x = 2x + 14##.

4. ##Simplifying gives: 7x = 14##

5. ##Finally, solving gives: x = 2##

Definition of Chord

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

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

Definition of Circle

<note>Let the center of the circle in the figure be O</note>, in circle O, <note>point O is the center of the circle</note>. <note>All points in the figure that are at a distance equal to the radius from point O lie on circle xxx</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>MN is the diameter</note>, connecting <note>the center O</note> and <note>points M and N on the circumference</note>, with a length of <note>twice the radius</note>.

Perpendicular Diameter Theorem

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

<note>Diameter MN is perpendicular to chord AB</note> and <note>chord DF</note>, then according to the Perpendicular Diameter Theorem, <note>diameter MN bisects chord AB and chord DF</note>, that is, <note>AC=BC</note>, <note>DE=FE</note>, and <note>diameter MN bisects the two arcs subtended by chord AB</note> and <note>chord DF</note>, that is, <note>arc AM = arc BM</note>, <note>arc DN = arc FN</note>.


Question: As shown in the figure, points A, B, C, P are on ⊙O, CD⊥OA, CE⊥OB, the feet of the perpendiculars are D and E respectively, ∠DCE=40°, then the degree of ∠P is ()

Answer: 70°

1. Given points A, B, C, P are on circle O, CD is perpendicular to OA, CE is perpendicular to OB, with feet at D and E respectively.

2. ##In quadrilateral ODCE, according to the polygon interior angle sum theorem, the interior angle sum of quadrilateral ODCE is (4-2) × 180° = 360°, i.e., ∠DOE + ∠OEC + ∠ECD + ∠CDO = 360°.##

####

##3. Given ∠DCE=40°. Since CD is perpendicular to OA, ∠CDO=90°; similarly, CE is perpendicular to OB, so ∠OEC=90°.##

##4. Substituting the given conditions, we get ∠DOE = 360°-∠OEC -∠ECD -∠CDO = 360°-90°-40°-90° = 140°.##

##5. Arc ACB corresponds to ∠AOB and ∠P. According to the definition of central angle, ∠AOB is the central angle; according to the definition of inscribed angle, ∠P is the inscribed angle.##

##6. According to the inscribed angle theorem, ∠P=1/2 * ∠AOB=1/2 * 140°=70°##

####

##7##. Through the above reasoning, the final answer is 70°.

圆; 垂线; 圆周角; 圆心角; 直角三角形

Definition of Arc

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

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

Definition of Inscribed Angle

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

Definition of Central Angle

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

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

Inscribed Angle Theorem

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

In circle O, points A, P, and B are on the circle, the arc ACB corresponds to the central angle ∠AOB and the inscribed angle ∠P. According to the Inscribed Angle Theorem, ∠P equals half of the central angle ∠AOB corresponding to the arc ACB, that is ∠P = 1/2 ∠AOB.

Polygon Interior Angle Sum Theorem

The sum of the interior angles of a polygon is equal to (n - 2) * 180°, where n represents the number of sides of the polygon.

<note>Quadrilateral ODEC is a polygon with 4 sides, where 4 represents the number of sides of the polygon</note>. According to the Polygon Interior Angle Sum Theorem, <note>the sum of the interior angles of the polygon is equal to (4-2) × 180° = 360°</note>. Therefore, <note>∠DOE + ∠OEC + ∠ECD + ∠CDO = 360°</note>.


Question: Find the value of x in the figure. Round to the nearest tenth if necessary.

Answer: 3

1. Given that there is an isosceles trapezoid in the problem, with the upper base being 6x + 4, the midline being 5x + 10, the lower base being 10x - 2, and the left and right sides being equal. ##Let the four vertices of the isosceles trapezoid be ABCD, and the midline be EF.##

2. According to the ##Midline Theorem of trapezoid##, the length of the midline of an isosceles trapezoid is equal to half the sum of the lengths of the upper base and the lower base. That is, midline == (upper base + lower base) / 2.

3. Substitute the given conditions: 5x + 10 = [(6x + 4) + (10x - 2)] / 2.

4. Expand and simplify the equation: 5x + 10 = (6x + 4 + 10x - 2) / 2.

5. Further simplify the right side: 5x + 10 = (16x + 2) / 2.

6. Remove the denominator: 5x + 10 = 8x + 1.

7. Move all x terms to one side of the equation and constant terms to the other side: 5x - 8x = 1 - 10.

8. Combine like terms to get: -3x = -9.

9. Divide both sides by -3 to get: x = 3.

10. Substitute x = 3 into the three length formulas to check correctness: upper base 6x + 4 = 6##x##(3) + 4 = 22, midline 5x + 10 = 5##x##(3) + 10 = 25, lower base 10x - 2 = 10##x##(3) - 2 = 28. The midline 25 is indeed equal to half the sum of the upper base 22 and the lower base 28 (25 = (22 + 28) / 2 = 25).

Definition of Isosceles Trapezoid

A trapezoid is isosceles if and only if its non-parallel sides (legs) are congruent (∅).

In the figure of this problem, <note>isosceles trapezoid ABCD</note>, <note>sides AB and DC are parallel (i.e., the two parallel sides of the trapezoid)</note>, <note>sides AD and BC are the legs of the trapezoid</note>, and <note>sides AD are equal to sides BC (i.e., the two legs are equal)</note>. Therefore, <note>ABCD is an isosceles trapezoid</note>.

Median Line Theorem of Trapezoid

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

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