Question: As shown in the figure, ABCD is a cyclic quadrilateral of ⊙O, and ∠ABC=115°, then ∠AOC equals ()

Answer: 130°

1. Given that points A, B, C, D are the vertices of a cyclic quadrilateral inscribed in ⊙O, and ∠ABC equals 115°.

2. ##According to (Corollary 3 of the Inscribed Angle Theorem) the theorem of supplementary angles in a cyclic quadrilateral##, i.e., ∠ABC + ∠ADC = 180°, thus ∠ADC = 180° - ∠ABC = 180° - 115° = 65°.

3. ##Arc ABC corresponds to ∠AOC and ∠ADC. According to the definition of central angle, ∠AOC is a central angle; according to the definition of inscribed angle, ∠ADC is an inscribed angle. According to the Inscribed Angle Theorem, we have ∠AOC = 2∠ADC.##

4. Substituting the known values, we get ∠AOC = 2 × 65° = 130°.

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

圆内接四边形; 圆周角; 圆心角; 圆

Definition of Arc

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

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

Cyclic Quadrilateral

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

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

Definition of Inscribed Angle

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

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

Definition of Central Angle

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

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

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

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

<note>Quadrilateral ABCD is a cyclic quadrilateral inscribed in circle O</note>, conforming to the theorem, thus <note>∠ABC + ∠ADC = 180°</note>. Given <note>∠ABC = 115°</note>, therefore <note>∠ADC = 180° - 115° = 65°</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>, points <note>A, B, C, and D are on the circle</note>, the central angle corresponding to arc <note>AC and arc BD</note> is <note>∠AOC</note>, and the inscribed angle is <note>∠ADC</note>. According to the <note>Inscribed Angle Theorem</note>, <note>∠AOC is equal to twice the inscribed angle ∠ADC corresponding to arc AC</note>, that is, <note>∠AOC = 2 × ∠ADC.</note>


Question: As shown in the figure, CD is a chord of ⊙O, O is the center of the circle. Fold the minor arc of ⊙O along CD, A is a point on the folded minor arc, ∠CAD=110°, then the degree of ∠B is ()

Answer: 70°

1. Given CD is a chord of ⊙O, ####∠CAD=110°。

2. ##Let A' be the symmetric point of A with respect to CD. Since folding the minor arc of ⊙O along CD, A is a point on the folded minor arc, so A' is on circle O##。

3. Auxiliary lines: connect A' and C, connect A' and D。

4. ##According to symmetry, △A'CD and △ACD are symmetric figures, so ∠CA'D =∠CAD = 110°##。

5. ##Since points A', B, C, D are all on circle O, according to the cyclic quadrilateral##, quadrilateral A'CBD is a cyclic quadrilateral of ⊙O。

6. According to ##(the corollary of the inscribed angle theorem 3) the opposite angles of a cyclic quadrilateral are supplementary, ∠CBD + ∠CA'D = 180°##。

8. Through the above reasoning, we finally obtain the degree of ##∠CBD, i.e., ∠B## is 70°。

圆; 弦; 圆周角; 对称; 反射

Symmetry

Symmetry refers to a geometric figure or pattern remaining invariant under certain operations such as rotation, reflection, or translation.

<note>Let point A's symmetric point with respect to CD be A'</note>, △ACD is symmetric with respect to <note>point CD</note>, so <note>CD is the center of symmetry of △ACD</note>. Each part of △ACD has a corresponding symmetric part on the other side of the center of symmetry. Therefore, <note>∠CA'D = ∠CAD = 110°</note>.

In the diagram of this problem, <note>quadrilateral A'CBD</note> is a <note>cyclic quadrilateral of circle O</note>. According to the <note>Diagonal Supplementary Theorem for Cyclic Quadrilateral</note>, the sum of each pair of opposite angles of <note>quadrilateral A'CBD</note> is equal to 180°. Specifically, <note>∠CBD + ∠CA'D = 180°</note>. A <note>cyclic quadrilateral</note> refers to a quadrilateral whose all four vertices lie on the same circle. This circle is called the <note>circumcircle</note> of the quadrilateral. In the diagram of this problem, <note>the four vertices A', B, C, D of quadrilateral A'CBD all lie on the same circle</note>. This circle is called the <note>circumcircle of quadrilateral A'CBD</note>. Therefore, <note>quadrilateral A'CBD is a cyclic quadrilateral</note>.


Question: Find x.

Answer: 6

1. There are two intersecting chords within a circle. Let one chord be AB and the other chord be CD. The chords AB and CD intersect at point P. PA = x, PB = x, PC = 4, PD = 9.

2. According to the intersecting chords theorem, if two chords intersect within a circle, then the products of their segments are equal, specifically PA * PB = PC * PD.

3. In this problem, we substitute the given conditions into the intersecting chords theorem formula: PA = x, PB = x, PC = 4, PD = 9, obtaining x * x = 4 * 9.

4. Calculating the equation 4 * 9, we get the right side of the equation as 36, thus the equation becomes x^2 = 36.

5. Solving the equation x^2 = 36, we get x = √36 or x = -√36.

6. Since x represents a length, the negative value has no physical meaning, so x = √36 = 6.

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

Definition of Chord

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

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

Intersecting Chords Theorem

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

In the <note>circle</note>, two chords <note>AB</note> and <note>CD intersect at point P</note>. <note>Chord AB is divided into two segments AP and PB by point P</note>, and <note>chord CD is divided into two segments CP and PD by point P</note>. According to the Intersecting Chords Theorem, it can be concluded that <note>the length of segment AP multiplied by the length of segment PB equals the length of segment CP multiplied by the length of segment PD</note>.


Question: As shown in the figure, AB // CD, EF intersects AB and CD at points E and F respectively, ∠1 = 50°, then the measure of ∠2 is ()

Answer: 130°

1. Given AB is parallel to CD, and ##∠1## = 50°.

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

3. Since ∠2 and ##∠CFE## are same-side interior angles, according to ##Parallel Postulate 2 of parallel lines, same-side interior angles are supplementary, it can be known## their sum is 180°, i.e., ∠2 + ##∠CFE## = 180°.

4. Substitute the given ##∠CFE## = 50°, we get ∠2 + 50° = 180°.

5. By calculation, we get ∠2 = 180° - 50° = 130°.

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

Definition of Parallel Lines

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

In the figure of this problem, <note>line AB and line CD lie 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>.

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>AB平行于CD，被直线EF所截</note>。##<note>∠CFE和∠2是同旁内角</note>，根据同旁内角互补定理，<note>∠CFE + ∠AEF = 180°</note>。已知<note>∠CFE = 50°</note>，所以<note>∠AEF = 180° - 50° = 130°</note>。##

Definition of Vertical Angles

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

<note>Two intersecting lines CD and EF intersect at point F</note>. According to the definition of vertical angles, <note>angle 1 and angle CFE are vertical angles</note>. Since the angles of vertical angles are equal, <note>angle 1 = angle CFE</note>.

Definition of Consecutive Interior Angles

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

<note>Two lines AB and CD</note> are intersected by a third <note>line EF</note>, the two <note>angles AEF and CFE</note> are on the same side of the intersecting line EF and within the intersected lines AB and CD, so <note>angles AEF and CFE are consecutive interior angles</note>. Consecutive interior angles AEF and CFE are supplementary, i.e., <note>angle AEF + angle CFE = 180 degrees</note>.


Question: Quadrilateral W X Y Z is a rectangle. If m \angle 1 = 30, find the degree measure of \angle 5.

Answer: 30

1. The given condition is that quadrilateral WXYZ is a rectangle, ##and angle 1 = 30°##.

2. ##According to the definition of a rectangle, WX and YZ are parallel, based on the parallel postulate 2 of parallel lines, angle 1 = angle 5##.

3. ##Through the above reasoning, the final answer is angle 5 = 30°##.

####

矩形; 对顶角; 邻补角

Definition of Rectangle

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

<note>Quadrilateral WXYZ is a rectangle</note>, <note>side WX is parallel to and equal in length to side YZ</note>, <note>side WY is parallel to and equal in length to side XZ</note>.

Parallel Postulate 2 of Parallel Lines

<note>Two parallel lines WX and YZ</note> are intersected by <note>a third line YX</note>, forming the following geometric relationship: <note>alternate interior angles: angle 1 = angle 5</note>. These relationships indicate that when two parallel lines are intersected by a third line, the alternate interior angles are equal.

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 WX and YZ</note> are intersected by <note>a third line YX</note>, where <note>angle 1 and angle 5 are between the two parallel lines</note> and on <note>opposite sides of the intersecting line YX</note>, thus <note>angle 1 and angle 5 are alternate interior angles</note>. <note>Alternate interior angles are equal, i.e., angle 1 = angle 5</note>.


Question: As shown in the figure, the lines AB and CD are intersected by the line EF. If AB∥CD and ∠1=100°, then the measure of ∠2 is ()

Answer: 80°

1. Given that line AB is parallel to line CD, let line EF intersect line AB and line CD at points G and H respectively.

2. According to the given condition, ∠1 = 100°, based on the definition of a straight angle, ∠1 + ∠AGH = 100° + ∠AGH = 180°, thus ∠AGH = 180° - 100° = 80°.

3. According to Parallel Postulate 2, corresponding angles are equal, therefore: ∠2 = ∠AGH = 80°.

4. Through the above reasoning, the final answer is ∠2 = 80°.

Definition of Parallel Lines

The lines <note>AB and CD</note> lie in the same plane and do not intersect, so according to the <note>definition of parallel lines</note>, <note>the lines AB and CD are parallel lines</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 GA rotates around endpoint G to form a straight line with the initial side, creating straight angle AGB</note>. According to the definition of a straight angle, <note>a straight angle measures 180 degrees, i.e., angle AGB = 180 degrees</note>.

Parallel Postulate 2 of Parallel Lines

<note>Line AB</note> and <note>Line CD</note> are intersected by <note>Line EF</note>, forming the following geometric relationships:
1. Corresponding angles: <note>∠AGH</note> and <note>∠CHF(∠2)</note> are equal.
2. Alternate interior angles: <note>∠BGH(∠1) and ∠CHG</note> are equal.
3. Consecutive interior angles: <note>∠HGC and ∠CHG</note> are supplementary, i.e., <note>∠HGC + ∠CHG</note> = 180 degrees.

These relationships illustrate that when two parallel lines are intersected by a third line, corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary.

Definition of Corresponding Angles

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

<note>Two parallel lines AB and CD</note> are intersected by a line <note>EF</note>, where <note>angle AGH</note> and <note>angle 1</note> are on <note>the same side of the transversal EF</note>, on the <note>same side of the intersected lines AB and CD</note>. Therefore, <note>angle AGH</note> and <note>angle 2</note> are corresponding angles. Corresponding angles are equal, that is, <note>angle AGH is equal to angle 2</note>.


Question: Using parallelogram ABCD, find m \angle AFD.

Answer: 97

####

##1##. In triangle AFD, it is known that ##angle FAD## = 49° and angle ADF = 34°. According to the triangle angle sum theorem, the sum of the interior angles in triangle AFD is 180°, that is ##angle AFD + angle FAD + angle ADF =## angle AFD + 49° + 34° = 180°.

##2##. Solving the equation gives: angle AFD = 180° - 49° - 34°.

##3##. After calculation, angle AFD = 97°.

##4##. Therefore, through the above reasoning, the final answer is that the measure of angle AFD is 97°.

平行四边形; 对顶角; 三角形的外角

Triangle Angle Sum Theorem

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

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


Question: As shown in the figure, AB∥CD, point E is on the extension of CA. If ∠BAE=40°, then the measure of ∠ACD is ()

Answer: 140°

1. Given AB∥CD, point E is on the extension of CA, and ∠BAE = 40°.

2. ##Draw CF as the extension of DC, then from the figure we get AB∥DF. According to the parallel lines axiom 2, corresponding angles are equal, so ∠BAE = ∠ECF = 40°##.

3. ##According to the definition of a straight angle, ∠ACD = 180° - ∠ECF##.

4. ##Substitute the given conditions to obtain ∠ACD = 180° - 40° = 140°##.

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

平行线; 内错角; 对顶角

Definition of Parallel Lines

In the figure of this problem, <note>line AB and line CD</note> lie in the same plane and have no intersection points, so according to the definition of parallel lines, <note>line AB and line CD</note> are parallel lines; <note>line AB and line DF</note> lie in the same plane and have no intersection points, so according to the definition of parallel lines, <note>line AB and line DF</note> are parallel lines.

Definition of Straight Angle

In the diagram of this problem, <note>ray CD rotates around endpoint C to form a straight line with the initial side</note>, forming <note>straight angle DCF</note>. According to the definition of a straight angle, <note>a straight angle measures 180 degrees, i.e., angle DCF = 180 degrees</note>.

Parallel Postulate 2 of Parallel Lines

<note>In the diagram of this problem, two parallel lines AB and CD are intersected by a third line CE</note>, forming the following geometric relationship: Corresponding angles: <note>∠BAE and ∠ECF are equal</note>.

Definition of Corresponding Angles

<note>Two parallel lines AB and CD are intersected by a transversal CE</note>, where <note>angle BAE and angle ECF are on the same side of the transversal CE and on the same side of the intersected lines AB and CD</note>, therefore <note>angle BAE and angle ECF are corresponding angles</note>. Corresponding angles are equal, that is, <note>angle BAE is equal to angle ECF</note>.


Question: As shown in the figure, AB is the diameter of ⊙O, points C and D are on ⊙O, and point C is the midpoint of arc BD. Through point C, a perpendicular line EF is drawn to AD, intersecting line AD at point E. If the radius of ⊙O is 2.5 and the length of AC is 4, then the length of CE is ()

Answer: \frac{12}{5}

1. Given AB is the diameter of ⊙O, AC is 4, the radius of ⊙O is 2.5, connect BC.

2. According to (Circle Angle Theorem Corollary 2) the angle subtended by the diameter is a right angle, ∠ACB=90°.

3. According to the definition of a right triangle, ∠ACB=90°, so △ABC is a right triangle.

4. Given the radius of the circle is 2.5, so the diameter AB=5, AC=4, according to the Pythagorean theorem, BC^2=AB^2-AC^2=5^2-4^2=25-16=9, so BC=3.

5. Since EF⊥AE, according to the definition of perpendicular lines, the angle AEC=90°. Point C is the midpoint of arc BD, so arc DC=arc BC, resulting in ∠DAC=∠CAB, ∠DAC=∠EAC. According to the similarity theorem (AA), △EAC∽△CAB.

6. According to the definition of similar triangles, ∴AC/AB=EC/BC.

9. Through the above reasoning, the final answer is 2.4.

圆; 直角三角形; 垂线; 弧; 圆周角

Definition of Inscribed Angle

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

Definition of Diameter

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

<note>AB is the diameter, connecting the center O and points A and B on the circumference</note>, with a length of 2 times the radius, i.e., AB = 2×2.5 = 5.

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 circle O, point O is the center of the circle, and points A and B are any points on the circle, the line segments OA and OB are the segments from the center to any point on the circle, therefore the line segments OA and OB are the radii of the circle, and OA=OB=2.5.

Definition of Perpendicular Lines

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

The original text: <note>The angle ∠AEC formed by the intersection of line EF and line AE is 90 degrees</note>, therefore, according to the definition of perpendicular lines, <note>line EF and line AE are perpendicular to each other</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 diagram of this problem, in circle O, the angle subtended by the diameter AB at the circumference ##∠ACB## is a right angle (90 degrees)</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 EAC and triangle CAB are similar triangles</note>. According to the definition of similar triangles: <note>∠EAC = ∠CAB</note>, <note>∠AEC = ∠BCA</note>, <note>∠ACE = ∠CBA</note>; <note>\(\frac{AC}{AB} = \frac{EC}{BC}\)</note>.

Pythagorean Theorem

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

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


Question: Find m \angle D.

Answer: 80

1. Quadrilateral ABCD is a cyclic quadrilateral in a circle, therefore according to the Opposite Angles of a Cyclic Quadrilateral Theorem.

4. Substituting the given angle expressions into the opposite angles relationship, we get: (3y + 4)° + (2y + 16)° = 180°.

5. Simplifying the equation, we get (5y + 20)° = 180°.

6. By removing the constant term, we get (5y)° = 160°.

8. After determining the value of y, substituting into ∠ADC = (2y + 16)°, we get ∠ADC = (2*32 + 16)° = (64 + 16)° = 80°.

9. Through the above reasoning, the final answer is ∠ADC = 80°.

圆内接四边形; 圆周角; 三角形的外角

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

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

Cyclic Quadrilateral

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

In the figure of this problem, the four vertices A, B, C, and D of quadrilateral ABCD are all on the same circle<note>在本题图中，四边形ABCD的四个顶点A、B、C和D都在同一个圆上</note>. This circle is called the circumcircle of quadrilateral ABCD<note>这个圆称为四边形ABCD的外接圆</note>. Therefore, quadrilateral ABCD is a cyclic quadrilateral. According to the properties of cyclic quadrilaterals, it can be concluded that<note>对角相加等于180度，即角ABC + 角ADC = 180度，角BAD + 角BCD = 180度</note>.


Question: Find x. Round to the nearest tenth.

Answer: 30.7

1. Given a right triangle ABC, where angle B is a right angle, the side opposite angle A is side BC, and the length of side AB is 11.

2. Angle C is 21°, so in the right triangle, the side opposite angle ACB is 11, and the hypotenuse is x. According to the definition of the sine function, we can use the sine function from trigonometry, since the definition of the sine function is: sin(θ) = opposite side/hypotenuse.

3. For angle C, sin(21°) = opposite side AB/hypotenuse AC, thus sin(21°) = 11/x.

4. Look up or calculate sin(21°) ≈ 0.3584.

5. Substitute the value of sin(21°) into the equation, yielding: 0.3584 = 11/x.

6. Solve the equation to get x = 11/0.3584 ≈ 30.691.

7. Round the result to the nearest tenth, obtaining x ≈ 30.7.

8. Through the above reasoning, the final answer is 30.7.

Definition of Sine Function

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

In the right triangle ABC, ∠ACB is an acute angle, <note>side AB is the opposite side of ∠ACB</note>, and side AC is the hypotenuse. According to the definition of the sine function, the sine value of ∠ACB is equal to the ratio of the opposite side <note>AB</note> to the hypotenuse AC, that is, sin(21°) = <note>AB</note> / AC.

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


Question: As shown in the figure, in △ABC, ∠C=36°, △ABC is rotated counterclockwise around point A by 60° to obtain △AED, AD intersects BC at point F, then the degree of ∠AFC is ()

Answer: 84°

1. Given ∠C = 36°, and △ABC is rotated counterclockwise around point A by 60° to obtain △AED.

2. Therefore, ∠DAC=60°, point F is on side AD, so ∠FAC=60°.

3. According to the triangle angle sum theorem, the sum of the three interior angles of any triangle is 180°, so ∠C+∠FAC+∠AFC=180°.

4. Given ∠C = 36°, ∠FAC=60°, so ∠AFC=180°-∠C-∠FAC=180°-36°-60°=84°.

5. Through the above reasoning, the final answer is 84°.

旋转; 点; 普通三角形; 内错角; 对顶角

Triangle Angle Sum Theorem

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

Definition of Rotation

In a plane, the transformation of rotating a figure about a point O by a certain angle is called rotation. Point O is referred to as the center of rotation, and the angle of rotation is called the angle of rotation. If a point P on the figure is transformed to point P' via the rotation, then these two points are called corresponding points of the rotation.

<note>Triangle DAE is obtained by rotating triangle CAB counterclockwise by 60° around point A</note>. Among them, <note>point D and point C are corresponding points</note>, <note>line segment AD and AC are corresponding line segments</note>, <note>∠D and ∠C are corresponding angles</note>, <note>point A is the center of rotation</note>, <note>the measure of ∠DAC is called the angle of rotation</note>.


Question: As shown in the figure, △ODC is obtained by rotating △OAB clockwise by 30° around point O. If point D happens to fall on AB, and the degree of ∠AOC is 100°, then the degree of ∠DOB is ()

Answer: 40°

1. According to the problem, △ODC is obtained by rotating △OAB clockwise by 30° around point O. Based on the properties of rotation, △ODC and △OAB are congruent, and points A and D, points B and C are corresponding points respectively. Therefore, ∠AOD = ∠BOC = 30°.

2. Given ∠AOC = 100°, ∠AOC = ∠AOD + ∠DOB + ∠BOC

3. Substituting the known angles into the equation, we get: 100° = 30° + ∠DOB + 30°.

4. Through calculation, we obtain: ∠DOB = 100° - 30° - 30° = 40°.

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

旋转; 普通三角形; 普通四边形; 邻补角; 对顶角

Properties of Rotation

1. The distance between corresponding points and the center of rotation is equal.
2. The angle between the segments connecting corresponding points and the center of rotation is equal to the angle of rotation.
3. The figure before and after rotation is congruent (≅).

<note>In the figure of this problem, the shape OAB is rotated around the rotation center O</note> 30° <note>to obtain the shape ODC</note>. According to the properties of rotation: 1. <note>The corresponding points (such as point A and point D) are equidistant from the rotation center O</note>, that is, <note>OA = OD</note>; 2. <note>The angle between the lines connecting the corresponding points and the rotation center is equal to the rotation angle of 30°</note>, that is, <note>∠AOD = 30°</note>, <note>∠BOC = 30°</note>; 3. <note>The shape OAB before rotation is congruent to the shape ODC after rotation</note>, that is, <note>shape OAB ≅ shape ODC</note>.


Question: As shown in the figure, line a∥b, ∠2=35°, ∠3=40°, then the degree of ∠1 is ()

Answer: 105°

1. Let the third angle forming the triangle in the figure be ∠4. Given that line a is parallel to line b, according to the parallel line axiom 2 and the definition of corresponding angles, corresponding angles are equal, and ∠1 and ∠4 are corresponding angles, we get ∠1 = ∠4.

2. According to the given conditions ∠2 = 35°, ∠3 = 40°, based on the triangle angle sum theorem, the sum of the interior angles in a triangle is 180°, we get ∠4 = 180° - ∠2 - ∠3.

3. Substituting the given angles, we get ∠4 = 180° - 35° - 40° = 105°.

4. According to step 1, we get ∠1 = ∠4. Therefore, ∠1 = 105°.

5. Through the above reasoning, we finally get the answer as 105°.

平行线; 同旁内角; 三角形的外角

Definition of Parallel Lines

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

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

Parallel Postulate 2 of Parallel Lines

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

In the figure of this problem, <note>two parallel lines a and b are intersected by a third line</note>, forming the following geometric relationships:
1. <note>Corresponding angles: angle 1 and angle 4 are equal</note>.
2. Alternate interior angles: none.
3. Same-side interior angles: none.
These relationships illustrate that <note>when two parallel lines are intersected by a third line, corresponding angles are equal</note>.

Definition of Corresponding Angles

In the figure of this problem, <note>two parallel lines a and b</note> are intersected by a transversal, where <note>angle 1 and angle 4 are on the same side of the transversal</note>, <note>on the same side of the two intersected lines a and b</note>, therefore <note>angle 1 and angle 4 are corresponding angles</note>. Corresponding angles are equal, that is, <note>angle 1 is equal to angle 4</note>.

Triangle Angle Sum Theorem

<note>Angle ∠2, Angle ∠3, and Angle ∠4 are the three interior angles of the triangle</note>, according to the Triangle Angle Sum Theorem, <note>Angle ∠2 + Angle ∠3 + Angle ∠4 = 180°</note>.


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

Answer: \frac{3}{5}

1. According to the problem, it is known that ∠C in △ABC is a right angle. Based on the Pythagorean theorem, we get AB##^2##=AC##^2##+BC##^2##.

2. Substitute the given conditions AB=5 and AC=4 into the Pythagorean theorem, obtaining 5##^2##=4##^2##+BC##^2##.

3. Calculate to get 25=16+BC##^2##, thus BC##^2##=9, resulting in BC=3.

4. From the above calculation, we get BC=3.

5. In a right triangle, the definition of sinA is the ratio of the length of the opposite side to the length of the hypotenuse.

6. In this case, sinA=BC/AB.

7. Based on the above calculation results and given conditions, we get sinA=3/5.

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

直角三角形; 正弦; 线段

Definition of Right Triangle

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

Definition of Sine Function

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

Pythagorean Theorem

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


Question: Triangle E F G is an equilateral triangle, and E H bisects \angle E. Find m \angle 2.

Answer: 30

1. Given that △EFG is an equilateral triangle, according to the property of equilateral triangles, each angle is 60°, i.e., ∠EFG, ∠FEG, and ∠EGF are all 60°.

2. E H is the angle bisector of ∠GEF, according to the definition of angle bisector, we get ∠2 = (1/2)∠GEF.

3. Given ∠GEF = 60°, substituting into ∠2 = (1/2)∠GEF, we get ∠2 = (1/2) * 60° = 30°.

####

##4##. The final result is ∠2 = 30°.

等边三角形; 内错角; 对称

Definition of Equilateral Triangle

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

In the figure of this problem, triangle EFG is an equilateral triangle. <note>The lengths of sides EF, FG, and GE are equal</note>, and <note>the measures of angles EFG, FEG, and EGF are equal, each being 60°</note>.

Definition of Angle Bisector

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

In the figure of this problem, <note>the vertex of angle GEF is point E</note>, a line EH is drawn from <note>point E</note>, <note>this line divides angle GEF into two equal angles, that is, angle GEH and angle HEF are equal</note>. Therefore, <note>line EH is the angle bisector of angle GEF</note>.

Angle Property of Equilateral Triangle

Each interior angle of an equilateral triangle is 60°.

In the figure of this problem, in the equilateral triangle EFG, <note>sides EF, FG, and GE are equal</note>, therefore, according to the properties of an equilateral triangle, <note>each interior angle of triangle EFG is 60°</note>. That is to say, <note>angle EFG, angle FEG, and angle EGF are all 60°</note>.


Question: As shown in the figure, quadrilateral ABCD is an inscribed quadrilateral of ⊙O, AB is the diameter of ⊙O, and BD is connected. If ∠BCD=120°, then the measure of ∠ABD is ()

Answer: 30°

1. Given that quadrilateral ABCD is a cyclic quadrilateral of ⊙O, according to the theorem ##(corollary 3 of the inscribed angle theorem) the opposite angles of a cyclic quadrilateral are supplementary##, we get ∠BAD + ∠BCD = 180°.

2. Since it is known that ∠BCD=120°, we can calculate ∠BAD=180°-120°=60°.

3. Also, since AB is the diameter of ⊙O, according to the theorem ##(corollary 2 of the inscribed angle theorem) the inscribed angle subtended by the diameter is a right angle##, we get ∠ADB=90°.

4. From the above conclusions, we can use the triangle angle sum theorem in △ABD to get, ##∠BAD + ∠ADB + ∠ABD = 60° +90° + ∠ABD° =180°##, ∠ABD=180°-∠ADB-∠BAD=180°-90°-60°=30°.

5. After the above reasoning, the final answer is 30°.

圆内接四边形; 圆周角; 直角三角形; 圆; 弦

Cyclic Quadrilateral

<note>The four vertices A, B, C, and D of quadrilateral ABCD lie on the same circle</note>. This circle is called <note>the circumcircle of quadrilateral ABCD</note>. Therefore, <note>quadrilateral ABCD is a cyclic quadrilateral</note>. According to the properties of a cyclic quadrilateral, it can be concluded that <note>the sum of opposite angles is equal to 180 degrees</note>, that is, <note>angle BAD + angle BCD = 180 degrees</note>, <note>angle ABC + angle ADC = 180 degrees</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>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>2 times the radius</note>, that is, <note>AB = 2 * r</note>.

Definition of Inscribed Angle

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

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

<note>Quadrilateral ABCD is a cyclic quadrilateral</note>. According to the Diagonal Supplementary Theorem for Cyclic Quadrilateral, <note>the sum of each pair of opposite angles of quadrilateral ABCD is equal to 180°</note>. Specifically, <note>∠BAD + ∠BCD = 180°</note>, <note>∠ABC + ∠ADC = 180°</note>. <note>Given that ∠BCD = 120°</note>, therefore <note>∠BAD = 180° - 120° = 60°</note>.

The angle subtended by <note>diameter AB</note> is a right angle (<note>90 degrees</note>).

Triangle Angle Sum Theorem

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

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


Question: A sector of paper with a central angle of 120° and a radius of 3cm is rolled into a bottomless cone (as shown in the figure). What is the height of this cone?

Answer: 2√{2}cm

1. Given a sector of paper with a central angle of 120° and a radius of 3 cm, it is rolled into a bottomless conical paper hat.

2. Let the radius of the base of the cone be x cm. According to the relationship of the circumference, we get the equation: 2πx = (120/360) * 2π * 3.

4. Since the radius of the sector forming the cone is the slant height of the cone, which is 3 cm, we can use the Pythagorean theorem to calculate the height h.

5. In the right triangle, the slant height is 3 cm, the radius of the base is 1 cm, and the height h is unknown. According to the Pythagorean theorem a^2 + b^2 = c^2.

6. Here a = h, b = 1, c = 3. Substituting, we get h^2 + 1^2 = 3^2.

8. Further calculation gives h^2 = 9 - 1 = 8, so h = √{8} = 2√{2}.

9. Through the above reasoning, the final answer is 2√{2} cm.

扇形; 圆锥; 圆心角; 直角三角形

Definition of Central Angle

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

<note>The central angle of the sector-shaped paper is 120°</note>. This <note>central angle</note> is an angle of the sector forming the cone, <note>with the center as the center point of the sector</note>.

Definition of Sector

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

In the figure of this problem, <note>the radius of 3 cm and the radius of 3 cm are two radii of the circle</note>, <note>the arc is the arc enclosed by these two radii</note>, so according to the definition of sector, the figure formed by <note>these two radii and the arc they enclose is a sector</note>.

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, in the cone, <note>the base is a circle</note>, the circle's <note>radius is 1 cm</note>. <note>The vertex of the cone is the vertex of the cone</note>, the distance between the vertex and <note>the center of the circle is the height of the cone</note>, denoted as h. <note>The lateral surface of the cone is a curved surface</note>, the distance from the vertex to <note>any point on the circumference is the slant height</note>, denoted as 3 cm.

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 \( r \): \( L = \theta r \).

Original: <note>In the diagram of this problem, the central angle of the sector is 120°, and the radius is 3 cm</note>. According to the formula for the length of an arc of a sector, <note>the arc length L is equal to the central angle θ multiplied by the radius r, i.e., L = θ * r</note>. <note>Given that the central angle is 120°, i.e., θ = 120° = 2π/3 (radians), and the radius r = 3 cm</note>, therefore <note>the arc length L = (2π/3) * 3 = 2π cm</note>. This <note>arc length is the circumference of the base of the cone</note>, so <note>2πx = 2π</note>, solving for <note>the base radius x = 1 cm</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>The slant height of the cone is 3 cm</note>, <note>the radius of the base is 1 cm</note>, using the Pythagorean Theorem to find <note>height h</note>. Let the height be h, applying the Pythagorean Theorem, <note>a is the height h</note>, <note>b is the radius of the base 1 cm</note>, <note>c is the slant height 3 cm</note>, we get h^2 + 1^2 = 3^2. Then solving, h^2 = 9 - 1 = 8, finally <note>h = √{8} = 2√{2}</note>.

Generatrix

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

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


Question: As shown in the figure, it is known that O is a point inside quadrilateral ABCD, OA=OB=OC, ∠ABC=∠ADC=65°, then ∠DAO+∠DCO=()

Answer: 165°

1. Given OA=OB=OC, according to the definition of an isosceles triangle, triangles OAB and OBC are both isosceles triangles. Therefore, ∠OAB=∠OBA and ∠OBC=∠OCB.

2. Given ∠ABC=65°, and ∠ABC=∠OBA+∠OBC, since ∠OAB=∠OBA and ∠OBC=∠OCB, we have ∠ABC=∠OAB+∠OCB=65°.

3. 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., ∠BAD+∠ADC+∠DCB+∠ABC=360°. Given ∠ABC=∠ADC=65°, so ∠BAD+∠DCB=360°-∠ABC-∠ADC=360°-65°-65°=230°.

4. Since ∠OAB+∠OCB=65°, ∠BAD=∠OAB+∠OAD, ∠DCB=∠OCB+∠OCD.

5. Therefore, ∠OAD=∠BAD-∠OAB, ∠OCD=∠DCB-∠OCB. Then ∠DAO+∠DCO=∠BAD-∠OAB+∠DCB-∠OCB=(∠BAD+∠DCB)-(∠OAB+∠OCB).

6. Given ∠BAD+∠DCB=230°, ∠OAB+∠OCB=65°, substituting into ∠DAO+∠DCO=(∠BAD+∠DCB)-(∠OAB+∠OCB) we get: ∠DAO+∠DCO=230°-65°=165°.

7. After the above reasoning, the final answer is 165°.

等腰三角形; 圆内接四边形; 普通四边形; 圆周角; 直线

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 OAB, <note>side OA and side OB are equal</note>, therefore <note>triangle OAB is an isosceles triangle</note>. Similarly, in triangle OBC, <note>side OB and side OC are equal</note>, therefore <note>triangle OBC is also an isosceles triangle</note>.

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

In <note>quadrilateral ABCD</note>, <note>angle ABC</note>, <note>angle BCD</note>, <note>angle CDA</note>, and <note>angle DAB</note> are the four interior angles of the quadrilateral. According to the <note>Sum of Interior Angles of a Quadrilateral Theorem</note>, the sum of these four interior angles is 360°, that is, <note>angle ABC + angle BCD + angle CDA + angle DAB = 360°</note>.

Properties of Isosceles Triangle

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

<note>In the figure of this problem, in the isosceles triangle OAB, sides OA and OB are equal</note>. Therefore, according to the properties of an isosceles triangle, <note>the angles opposite the equal sides are equal, that is, angle OAB = angle OBA</note>. Similarly, <note>in the isosceles triangle OBC, sides OC and OB are equal</note>. Therefore, according to the properties of an isosceles triangle, <note>the angles opposite the equal sides are equal, that is, angle OCB = angle OBC</note>.


Question:
In the figure above, 3 < a < 5 and 6 < b < 8. Which of the following represents all possible values of c?

(A) 0 < c < 3
(B) 1 < c < 3
(C) 0 < c < 13
(D) 1 < c < 13
(E) 3 < c < 13

Answer: 1 < c < 13

1. Given the inequalities 3 < a < 5 and 6 < b < 8, according to the Triangle Inequality Theorem, any side length must be less than the sum of the other two side lengths and greater than the difference of the other two sides.

2. According to the Triangle Inequality Theorem, side c should be greater than the minimum value of b - a, which is obviously greater than 1. Similarly, side c should be less than the maximum value of b + a, which is obviously less than 13.

3. Therefore, the range of c should be 1 < c < 13.

Definition of Triangle

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

In the figure of this problem, a triangle is a geometric shape formed by three non-collinear points and their connecting line segments. <note>a, b, and c are the three sides of the triangle</note>. <note>a and b are known sides</note>, within their intervals <note>3 < a < 5 and 6 < b < 8</note>, <note>c represents the third side</note>.

Theorem of Triangle Inequality

In any triangle, the sum of the lengths of any two sides is greater than the length of the third side, and the absolute difference of the lengths of any two sides is less than the length of the third side.

<note>Side a, side b, and side c form a triangle</note>. According to the theorem of triangle inequality, the sum of any two sides is greater than the third side, that is, <note>side a + side b > side c</note>, <note>side a + side c > side b</note>, <note>side b + side c > side a</note>. The difference of any two sides is less than the third side, that is, <note>side a - side b < side c</note>, <note>side a - side c < side b</note>, <note>side b - side c < side a</note>.


Question: Find m \angle H.

Answer: 97

1. Given pentagon EFGHJ, its interior angles are: ∠E = x°, ∠F = (x + 20)°, ∠G = (x + 5)°, ∠H = (x - 5)°, ∠J = (x + 10)°.

2. ##According to the polygon interior angle sum theorem##, the sum of the interior angles of a pentagon is: (n - 2) * 180°, where n is the number of sides. In this problem, n = 5, so the sum of the interior angles of the pentagon is: (5 - 2) * 180° = 3 * 180° = 540°.

3. Set up the equation to solve for the interior angle sum equal to 540°: x + (x + 20) + (x + 5) + (x - 5) + (x + 10) = 540.

4. Combine like terms to get: 5x + 30 = 540.

5. Solve this equation: 5x + 30 = 540, to get: 5x = 510, thus x = 102.

6. Obtain the values of each angle: ∠E = 102°, ∠F = 122°, ∠G = 107°, ∠H = 97°, ∠J = 112°.

7. Through the above reasoning, the final answer is 97°.

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.

In the polygon <note>EFGHJ</note>, <note>EFGHJ is a polygon with n sides</note>, where <note>n represents the number of sides of the polygon</note>, <note>n=5</note>. The <note>sum of the interior angles of the pentagon equals (5 - 2) * 180° = 540°</note>.


Question: As shown in the figure, there is a sector with a central angle of 120° and a radius of 6cm. If OA and OB are overlapped to form the lateral surface of a cone, then the diameter of the base of the cone is ()

Answer: 4cm

1. Given that the central angle of the sector is 120°, and the radius is 6cm. ##According to the conversion formula between degrees and radians, convert the central angle into radians to get θ = 120° × (π/180°) = (2π/3). According to the formula for the arc length of a sector L = θr (where L is the arc length, θ is the central angle in radians, and r is the radius), the arc length L = (2π/3) × 6 = 4π cm##.

2. Since OA and OB coincide to form the lateral surface of the cone, ##according to the unfolded diagram of the cone##, the circumference of the cone's base is equal to the arc length of the sector, which is 4π cm.

3. Let the radius of the cone's base be r, then the formula for the circumference of the cone's base is C = 2πr.

4. Based on the given conditions, we get the equation 2πr = 4π, solving for r gives r = 2 cm.

5. ##Let## the diameter of the cone's base ##be d##, ##according to the definition of diameter##, d = 2r = 2 × 2 = 4 cm.

6. Through the above reasoning, the final answer is 4 cm.

扇形; 圆心角; 圆锥; 圆; 弧

Definition of 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</note> is called the central angle, and the central angle measures 120°.

Definition of Sector

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

Cone

<note>The base of the cone is a circle</note>, the radius of the circle is <note>2 cm</note>, and the <note>center of the circle is the center of the base</note>. The vertex of the cone is <note>point O</note>, and the distance between the vertex O and the <note>center of the circle is the height of the cone</note>. <note>The lateral surface of the cone is a curved surface</note>, and the distance from the vertex O to <note>any point on the circumference is 6 cm</note>.

In the figure of this problem, in the sector <note>OAB</note>, <note>the central angle ∠AOB is 120°, which is (2π/3) in radians</note>, <note>the radius OA is 6 cm</note>. According to the formula for the length of an arc of a sector, <note>the arc length L equals the central angle θ multiplied by the radius r, i.e., L = θ * r</note>. Therefore, <note>the arc length L = (2π/3) × 6 = 4π cm</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 with radius r</note>. According to the <note>Circumference Formula of Circle</note>, <note>the circumference C of the circle is equal to 2π times the radius r, i.e., C=2πr</note>. Since <note>the sector OA and OB overlap to form the lateral surface of the cone</note>, <note>the circumference of the base of the cone is equal to the arc length of the sector, which is 4π cm</note>, thus <note>2πr = 4π</note>, solving for r gives <note>r = 2 cm</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, the development of the cone is a sector. <note>The radius of the sector is the slant height OA of the cone</note>, <note>the arc length of the sector is the circumference of the base circle of the cone</note>. Therefore, <note>the radius OA of the sector equals the slant height of the cone</note>, <note>the arc length AB of the sector equals the circumference of the base circle of the cone</note>, which is 2πr, where r is the radius of the base circle.

Formula for Conversion between Degrees and Radians

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

In the <note>figure OAB</note>, the <note>angle ∠AOB is 120°</note>. According to the formula for conversion between degrees and radians, it converts to <note>radians</note> = <note>degrees * (π/180)</note>, which is <note>120° * (π/180)</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>d is the diameter</note>, connecting the center of the circle and two points on the circumference, with a length equal to 2 times the radius, that is, <note>d = 2r</note>.


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

Answer: 54

1. Let the quadrilateral in the figure be quadrilateral ABCD (clockwise from the top left corner), with the extension of side DC towards point C being CE. From the arrows drawn on lines AB and CD in the figure, it indicates that these two lines are parallel.

2. According to the parallel axiom 2 and the definition of alternate interior angles, the angles located between two parallel lines and on opposite sides of the transversal are called alternate interior angles, and the measures of alternate interior angles are equal. Therefore, (2x)° = 108°.

3. Solve the equation 2x = 108, to get x = 108/2 = 54.

4. Through the above reasoning, the final answer is x = 54.

平行四边形; 内错角; 同旁内角

Definition of Quadrilateral

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

<note>Quadrilateral ABCD is composed of four line segments AB, BC, CD, and DA</note>, which are called the sides of the quadrilateral. Quadrilateral ABCD has four vertices, namely <note>point A, point B, point C, and point D</note>, and has four interior angles, namely <note>angle A, angle B, angle C, and angle D</note>.

Definition of Parallel Lines

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

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

Parallel Postulate 2 of Parallel Lines

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

In the figure of this problem, let the quadrilateral in the figure be quadrilateral ABCD, and the extension line of side DC towards point C be CE. <note>Two parallel lines AB and CD are intersected by a third line BC</note>, forming the following geometric relationships:

1. Corresponding angles: None.
2. Alternate interior angles: <note>Angle ABC and angle BCE are equal</note>.
3. Same-side interior angles: None.

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

Definition of Alternate Interior Angles

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

In the diagram of this problem, <note>two parallel lines AB and CD</note> are intersected by <note>a line BC</note>, where <note>angle ABC and angle BCE are located between the two parallel lines</note>, and <note>on opposite sides of the intersecting line BC</note>, therefore <note>angle ABC and angle BCE are alternate interior angles</note>. Alternate interior angles are equal, that is, <note>angle ABC is equal to angle BCE</note>.


Question: Quadrilateral ABCD is a rhombus. If AB = 2x + 3 and BC = x + 7, find CD.

Answer: 11

1. Given that quadrilateral ABCD is a rhombus, according to the definition of a rhombus, all sides are equal, therefore AB = BC = CD = DA.

2. Given AB = 2x + 3, BC = x + 7, we can obtain the equation 2x + 3 = x + 7.

3. By solving the equation 2x + 3 = x + 7, we get 2x - x = 7 - 3, which means x = 4.

4. Substituting x = 4 into BC = x + 7, we get BC = 4 + 7 = 11.

5. Since CD = BC, therefore CD = 11.

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

Definition of Rhombus

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

In the figure of this problem, the quadrilateral ABCD is a rhombus, so <note>the four sides AB, BC, CD, and DA are all equal</note>. <note>That is, AB = BC = CD = DA</note>.


Question: Lines l, m, and n are the perpendicular bisectors of triangle PQR and intersect at point T. If TQ = 2x, PT = 3y - 1, and TR = 8, find z.

Answer: 3

1. Let the intersection point of line l and PR be A. It is known that lines l, m, n are the perpendicular bisectors of △ PQR and they intersect at point T.

2. Since the distance from point P to point A is z + 4 and it is known that AR = 7.

3. According to the properties of perpendicular bisectors, AP = AR, therefore z + 4 = 7.

4. Solving this, we get z = 3.

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

垂直平分线; 直角三角形; 对称; 等腰三角形

Definition of Perpendicular Bisector

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

In the diagram of this problem, <note>line l passes through the midpoint A of segment PR</note>, and <note>line l is perpendicular to segment PR</note>. Therefore, <note>line l is the perpendicular bisector of segment PR</note>.


Question: a = 14, b = 48, and c = 50, find \sin A.

Answer: 0.28

1. Given that in triangle ABC, a = 14, b = 48, c = 50, and ∠C = 90°.

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

3. Based on the definition of the sine function, sinA is the ratio of the opposite side to the hypotenuse, sinA = opposite side/hypotenuse.

5. Through the above reasoning, the final answer is 0.28.

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

Definition of Sine Function

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

<note>Right triangle ABC</note><note>Angle ∠BAC is an acute angle</note><note>Side BC is the opposite side of angle ∠BAC</note><note>Side AC is the hypotenuse</note>According to the definition of the sine function, <note>the sine of angle ∠BAC is equal to the ratio of the opposite side BC to the hypotenuse AC</note>, that is, <note>sin(∠BAC) = BC / AC = a / c = 14 / 50</note>.


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

Answer: 207.8

1. Given the length of the base of the parallelogram is 12 cm, according to the height of the parallelogram, it is known that the two angles formed by the height and the base are both 90°. According to the definition of a right triangle, in this problem, the segment of the height forms a right triangle with a 60° angle and a 20 cm side. According to the triangle angle sum theorem, the other acute angle of this right triangle is 180° - 90° - 60° = 30°.

2. Let the hypotenuse of this right triangle be c (where c = 20 cm), the base be d, and the height be h. According to the definition of the sine function, sin(60°) = h / 20. Therefore, h = 20 * sin(60°).

3. According to the properties of a 30°-60°-90° triangle, sin(60°) = √3/2. Therefore, h = 20 * √3/2 = 20 * 0.8660254 = 17.32 cm.

4. The area calculation formula for a parallelogram is: base length multiplied by height (where the height is the vertical distance from the base to the opposite side). So, the area A = base length * height.

5. In this problem, A = 12 cm * 17.32 cm.

6. After calculation, A ≈ 207.8 (rounded to one decimal place).

7. Through the above reasoning, the final answer is 207.8.

Definition of Parallelogram

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

<note>A quadrilateral is a parallelogram</note>, <note>the base is 12 cm</note>, and <note>one side is 20 cm</note>. <note>Both pairs of opposite sides are parallel and equal</note>.

Definition of Right Triangle

<note>In a triangle, one angle is 90°</note>, another angle is 60°, therefore, the triangle is a right triangle.

Height of a Parallelogram

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

<note>In a parallelogram, the top and bottom sides are parallel</note>. <note>From the top vertex, draw a line segment perpendicular to the 12 cm side</note>, the length of this line segment is the height of the parallelogram.

Area Formula of a Parallelogram

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

In the figure of this problem, <note>the length of the base of the parallelogram is 12 cm</note>, and the corresponding height is the <note>vertical distance from the base to the opposite side, ##through calculation, the height is 17.32 cm##</note>. Therefore, according to the area formula of a parallelogram, <note>the area of the parallelogram is equal to the base length of 12 cm multiplied by the corresponding height of 17.32 cm, i.e., A = 12 cm × 17.32 cm</note>.

Definition of Sine Function

<note>In a right triangle, 60° is an acute angle</note>, <note>side h is the side opposite to the 60° angle</note>, <note>side 20 cm is the hypotenuse</note>. According to the definition of the sine function, the sine value of the 60° angle is equal to <note>the ratio of the opposite side h to the hypotenuse 20 cm</note>, that is, <note>sin(60°) = h / 20</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</note>, <note>the height h is the side opposite the 60° angle</note>, <note>the 20cm side is the hypotenuse</note>, according to the properties of the 30°-60°-90° triangle, let the short side be 1, then <note>h is √3</note>, <note>the hypotenuse is 2</note>, <note>then sin(60°) = √3/2</note>.

Triangle Angle Sum Theorem

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

In the figure of this problem, in the triangle formed by the 20 cm side, height h, and the base, according to the Triangle Angle Sum Theorem, <note>60° + 90° + another interior angle = 180°</note>, then <note>another interior angle is 30°</note>.


Question: Find m \angle R.

Answer: 58

1. Through understanding the quadrilateral and its angle relationships, the sum of the interior angles of any quadrilateral is 360° (quadrilateral interior angle sum theorem), i.e., ∠R + ∠Q + ∠S + ∠T = 360°.

3. Combining the angles, we get 2x + 5 + 2x + 7 = 4x + 12, then 2∠R + 4x + 12 = 360°.

4. Rearranging, we have 2∠R = 360 - 4x - 12, simplifying to 2∠R = 348 - 4x.

5. ∠R = 174 - 2x, now we need to solve for x using the given conditions.

6. Since ∠R and ∠T are equal and both are x°, we get the equation x = 174 - 2x.

7. Solving the equation x = 174 - 2x, we get 3x = 174, thus x = 58°.

8. Now we have ∠R = x = 58°, so the final angle ∠R is 58°.

Definition of Quadrilateral

The quadrilateral <note>QRST</note> is composed of four line segments <note>QR, RS, ST, and TQ</note>, which are referred to as the sides of the quadrilateral. The quadrilateral <note>QRST</note> has four vertices, namely <note>point Q, point R, point S, and point T</note>, and it has four interior angles, namely <note>angle Q, angle R, angle S, and angle T</note>.

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

In the figure of this problem, quadrilateral <note>RQTS</note> has <note>angles R, Q, S, and T</note> as its four interior angles. According to the Sum of Interior Angles of a Quadrilateral Theorem, the <note>sum of these four interior angles is 360°</note>, i.e., <note>∠R + ∠Q + ∠S + ∠T = 360°</note>.


Question: If line segment BC is the radius of the circle and line segment AB is tangent to the circle, what is the length of line segment AB in the figure?

Answer: 6*\sqrt{3}

1. Given that BC is the radius of circle C and AB is the tangent to circle C, by the property of the tangent to a circle, we have ∠CBA=90°.

2. Given ∠CAB=30°, combined with the conclusion from the previous step, according to the triangle angle sum theorem, we deduce ∠ACB=180°-∠CBA-∠CAB=180°-90°-30°=60°.

3. According to the definition of a right triangle, △ABC is a right triangle, and ∠ACB=30°. Based on the properties of a 30°-60°-90° triangle, AB is √3 times BC, i.e., AB=√3×BC.

4. Given that the length of BC is 6, from the previous calculation, AB=√3×6=6√3.

5. Through the above reasoning steps, we can conclude that the length of AB is 6√3.

切线; 直角三角形; 圆; 正弦

Definition of Right Triangle

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

Definition of Tangent to a Circle

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

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

Definition of Radius

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

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

Triangle Angle Sum Theorem

<note>In the given diagram, in triangle ABC, angle ABC, angle CBA, and angle ACB are the three interior angles of triangle ABC</note>, according to the Triangle Angle Sum Theorem, <note>angle ABC + angle CAB + angle ACB = 180°</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 C, point B is the tangent point of line AB with the circle, <note>line segment BC is the radius of the circle</note>. According to the property of the tangent line to a circle, <note>the tangent line AB is perpendicular to the radius BC passing through the tangent point B</note>, that is, <note>∠ABC=90°</note>.

Properties of 30°-60°-90° Triangle

<note>Triangle ABC is a 30°-60°-90° triangle</note>, where <note>∠CAB=30°</note>, <note>∠ACB=60°</note>, <note>∠ABC=90°</note>. <note>Side BC is opposite the 30-degree angle</note>, side AB is opposite the 60-degree angle, <note>side AC is the hypotenuse</note>. According to the properties of a 30°-60°-90° triangle, <note>side BC is equal to half of side AC</note>, side AB is equal to side BC multiplied by √3. That is: <note>BC = 1/2 * AC</note>, <note>AB = BC * √3</note>.


Question: Find the value of x. Round to the nearest tenth.

Answer: 30.2

1. Given conditions: ∠GDF = 125°, DG = 15, GF = 19. According to the cosine rule, we can use the following formula to solve for x (the length of DF): DF^2 = DG^2 + GF^2 - 2 * DG * GF * cos(∠GDF).

2. Substitute the given conditions into the formula: DF^2 = 15^2 + 19^2 - 2 * 15 * 19 * cos(125°).

3. Calculate the values of each term: 15^2 = 225, 19^2 = 361, 2 * 15 * 19 = 570, cos(125°) ≈ -0.5736.

4. Substitute these values into the formula: DF^2 = 225 + 361 - 570 * (-0.5736), i.e., DF^2 = 225 + 361 + 326.952 = 912.952.

5. Find the square root of DF: x = √912.952 ≈ 30.2.

6. Through the above reasoning, the final answer is 30.2.

Cosine Theorem

The cosine function is a trigonometric function defined in terms of 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.

<note>In the figure of this problem, in triangle △DGF, sides DG, GF, and DF correspond to the three sides of the triangle</note>, <note>angle ∠GDF is the angle formed between sides DG and GF</note>. According to the Cosine Theorem, <note>the square of side DF is equal to the sum of the squares of the other two sides DG and GF minus twice the product of these two sides and the cosine of the angle ∠GDF</note>, that is, <note>##DF? = DG? + GF?## - 2 * DG * GF * cos(∠GDF)</note>.


Question: As shown in the figure, AB is a chord of ⊙O, OC ⊥ AB, intersecting ⊙O at point C. Connect OA, OB, and BC. If ∠ABC = 25°, then the measure of ∠AOB is ()

Answer: 100°

1. Given that OC is perpendicular to AB, and OC intersects ⊙O at point C, ##suppose AB intersects OC at point D##, according to the perpendicular bisector theorem, OC bisects chord AB, that is, ##AD=BD##.

2. ##According to the definition of radius, OA=OB, OD is the common side, so according to the congruent triangle theorem (SSS), △AOD and △BOD are congruent, according to the definition of congruent triangles, ∠BOC=∠AOC##.

3. According to the problem, ∠ABC=25°.

4. ##According to the definition of inscribed angle, ∠ABC is an inscribed angle, according to the definition of central angle, ∠AOC is a central angle##, according to the inscribed angle theorem, ##∠ABC=?∠AOC, thus## ∠AOC=2∠ABC.

5. Substitute the value of ∠ABC, then ∠AOC=2×25°=50°.

6. ##According to the figure##, ∠AOB = ∠AOC + ∠BOC.

7. ##Since## ∠AOC = ∠BOC, then ∠AOB=2∠AOC.

8. Substitute the value of ∠AOC, then ∠AOB=2×50°=100°.

9. Through the above reasoning, the final answer is ∠AOB=100°.

圆; 圆周角; 圆心角; 垂线; 弦

Definition of Radius

In the diagram of this problem, <note>circle O</note>, <note>point O is the center of the circle</note>, <note>points A and B are any points on the circle</note>, <note>segments OA and OB are the segments from the center to any point on the circle</note>, therefore <note>segments OA and OB are the radii of the circle</note>, thus OA=OB##.

Perpendicular Diameter Theorem

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

In the figure of this problem, <note>AB intersects OC at point D</note>, in circle O, diameter OC is perpendicular to chord AB. According to the Perpendicular Diameter Theorem, diameter OC bisects chord AB, that is <note>AD = BD</note>, and diameter OC bisects the two arcs subtended by chord AB, that is <note>arc AC = arc CB</note>.

Triangle Congruence Theorem (SSS)

Two triangles are congruent if their three pairs of corresponding sides are equal in length.

<note>In the diagram of this problem, in triangle AOD and triangle BOD</note>, <note>side OA is equal to side OB</note>, <note>side AD is equal to side BD</note>, <note>side OD is equal to side OD</note>, therefore, according to the Triangle Congruence Theorem (SSS), <note>these two triangles are congruent</note>.

Definition of Congruent Triangles

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

<note>In the diagram of this problem, triangle AOD and triangle BOD are congruent triangles</note>, the corresponding sides and corresponding angles of triangle AOD are equal to those of triangle BOD, and the corresponding angles are also equal: <note>∠BOC=∠AOC</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, B, and C are on the circle</note>, the central angles corresponding to arc AC and arc BC are <note>∠AOC and ∠BOC</note>, and the inscribed angle is <note>∠ABC</note>. According to the Inscribed Angle Theorem, <note>∠ABC is half of the central angle ∠AOC corresponding to arc AC</note>, that is, <note>∠ABC = 1/2 ∠AOC</note>.

Definition of Central Angle

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

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

Definition of Inscribed Angle

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

In the figure of this problem, in circle O, the vertex of angle ABC (point B) is on the circumference, and the two sides of angle ABC intersect circle O at point A and point C respectively. Therefore, angle ABC is an inscribed angle.


Question: If A E = m - 2, E C = m + 4, A D = 4, and A B = 20, find m.

Answer: 4

1. In △ABC, DE∥BC, according to the parallel postulate 2 and the corresponding angles theorem, ∠AED=∠ACB. Also, ∠DAE=∠BAC, according to the similarity criterion (AA), △ADE∽△ABC.

2. Given AE = m - 2, EC = m + 4, AD = 4, AB = 20, according to the definition of similar triangles, AE/AC=AD/AB, and AC=AE+EC=(m-2)+(m+4)=2m+2.

3. Substituting the given conditions into the equation: (m - 2) / (2m + 2) = 4 / 20.

4. Solve the equation (m - 2) * 20 = (2m + 2) * 4.

5. Simplify the equation to get 20m - 40 = 8m + 8.

6. Rearrange and combine like terms to get 20m - 8m = 40 + 8.

线段; 三角形的外角; 普通三角形

Parallel Postulate 2 of Parallel Lines

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

In the figure of this problem, <note>two parallel lines DE and BC</note> are intersected by a third line <note>AC</note>, forming the following geometric relationships:
1. Corresponding angles: <note>angle AED and angle ACB are equal</note>.
These relationships illustrate that when two parallel lines are intersected by a third line, corresponding angles are equal, alternate interior angles are equal, and same-side interior angles are supplementary.

Definition of Corresponding Angles

Two angles are called corresponding angles if they lie on the same side of a transversal line c that intersects two other lines a and b, and each angle is on the same relative side of lines a and b.

In this problem diagram, <note>two parallel lines DE and BC are intersected by a line AC</note>, where <note>angle AED and angle ACB are on the same side of the transversal AC and on the same side of the intersected lines DE and BC</note>, thus <note>angle AED and angle ACB are corresponding angles</note>. <note>Corresponding angles are equal, that is, angle AED is equal to angle ACB</note>.

Similarity Theorem for Triangles (AA)

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

In the diagram of this problem, in triangles ADE and ABC, if <note>angle AE is equal to angle ACB</note>, and <note>angle DAE is equal to angle BAC</note>, then <note>triangle ADE is similar to triangle ABC</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>angle DAE = angle BAC</note>, <note>angle AED = angle ACB</note>, <note>angle ADE = angle ABC</note>; <note>AD/AB = AE/AC = DE/BC</note>.


Question: Find the length of \widehat J K. Round to the nearest hundredth.

Answer: 13.74

1. Given that the diameter of the circle is 15 cm, #### \angle JCK = 105°, and \angle JCK is the central angle. Therefore, ##arc JK## is the arc subtended by this central angle.

2. Based on the diameter of the circle being 15 cm, we can determine the radius r as half of the diameter, i.e., r = 15 / 2 = 7.5 cm.

3. According to the formula for the arc length of a sector: arc length l = r \theta, where \theta is the measure of the central angle in radians. We need to convert \angle JCK = 105° to radians, ##based on the conversion formula between degrees and radians: radians = degrees * (π/180), i.e., \theta = 105° * π/180##.

4. Calculate \theta = \theta = ##105° * π/180 = 7π/12##.

5. Using the formula for the arc length of a sector to calculate the length of arc JK: l = 7.5 * 7π/12##.

6. The theoretical calculation yields ##l = 7.5 * 7π/12 = 4.375π##.

7. Compute the above expression to obtain the approximate value ## l = 13.74406##.

8. After rounding, the final answer is ## l = 13.74 cm##.

Definition of Circle

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

In the figure of this problem, <note>in the circle, point C is the center</note>, <note>the radius is 7.5 cm</note>. In the figure, <note>all points that are 7.5 cm away from point C are on the circle</note>.

Definition of Arc

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

In the diagram of this problem, there are two points J and K on the circle, <note>arc JK</note> is a segment of the curve connecting these two points. According to the definition of an arc, <note>arc JK</note> is a segment of the curve between two points J and K on the circle.

Definition of Central Angle

In the figure of this problem, in the circle, <note>point J and point K are two points on the circle</note>, <note>the center of the circle is point C</note>. <note>The angle ∠JCK formed by the lines CJ and CK is called the central angle</note>.

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

In the figure of this problem, it is known that <note>the radius of the circle is 7.5 cm</note> and <note>\angle JCK = 105°</note>. We need to convert the angle to radians to use the arc length formula. <note>The formula for calculating the arc length JK is: l = 7.5 * (7π/12)</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>AJ is the diameter</note>, connecting <note>the center C</note> and <note>points A and J on the circumference</note>, with a length of 2 times the radius, that is, <note>AJ = 15 cm</note>.

Definition of Radius

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

<note>In circle C, point C is the center of the circle</note>, <note>point J is any point on the circle</note>, <note>line segment CJ is the line segment from the center to any point on the circle</note>, therefore <note>line segment CJ is the radius of the circle</note>.

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

In <note>Circle C</note>, <note>the angle ∠JCK is 105°</note>. According to the formula for conversion between degrees and radians, the conversion to radians is angle * (π/180), which is <note>105 * (π/180)</note>.


Question: Find the degree measure of \angle 2.

Answer: 76

1. We need to calculate the angle of ##∠2##.

2. First, using the triangle angle sum theorem, which states that the sum of the interior angles of a triangle is 180°.

3. Consider the first triangle, where two angles are known to be 47° and 57°.

4. Using the triangle angle sum theorem to calculate the third interior angle ∠1: 180° - 47° - 57° = 76°.

5. From the figure given in the problem, we observe that ##∠1 and ∠2 are vertical angles##.

6. ##According to the definition of vertical angles, we have ∠1 = ∠2##.

7. Therefore, the degree of ##∠2## is 76°.

8. Through the above reasoning, the final answer is 76°.

对顶角; 三角形的外角; 普通三角形

Triangle Angle Sum Theorem

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

In the figure of this problem, in the triangle, <note>angle 47°</note>, <note>angle 57°</note>, and <note>angle 1 are the three interior angles of the triangle</note>. According to the Triangle Angle Sum Theorem, <note>angle 47° + angle 57° + angle 1 = 180°</note>.

Definition of Vertical Angles

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

Original text: <note>Two angles: ∠1, ∠2</note>. According to the definition of vertical angles, <note>∠1 and ∠2 are vertical angles</note>. Since vertical angles are equal, <note>∠1 = ∠2</note>.


Question: Find m \angle 1.

Answer: 70

1. Given that angle 35 degrees, angle 75 degrees, and angle 1 in the ##figure## form a triangle. ##Let this triangle be △ABC,## according to the triangle angle sum theorem, the sum of the interior angles of a triangle is 180 degrees.

2. Based on the given conditions, we can write the equation: 35 degrees + 75 degrees + angle 1 = 180 degrees.

3. Solving this equation, we get: angle 1 = 180 degrees - 35 degrees - 75 degrees = 70 degrees.

####

##4##. Finally, through the above steps, the value of ##angle 1## is ##70 degrees##.

Definition of Triangle

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

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

Triangle Angle Sum Theorem

In the figure of this problem, in the triangle, <note>angle 35 degrees, angle 75 degrees, and angle 1</note> are the three interior angles of the triangle. According to the Triangle Angle Sum Theorem, <note>angle 35 degrees + angle 75 degrees + angle 1 = 180°</note>. Meanwhile, <note>another triangle containing angle 3 and angle 28 degrees</note> also follows this theorem, and its interior angles also sum up to 180 degrees.


Question: As shown in the figure, AB is a long ladder leaning against the wall, the foot of the ladder B is 1.6 meters away from the wall, the point D on the ladder is 1.4 meters away from the wall, BD is 0.55 meters long, then the length of the ladder is ()

Answer: 4.40米

1. Given that a ladder is leaning against a wall, let the top of the ladder be point A, the bottom of the ladder be point B, and AB be the length of the ladder. The distance from point B to the base of the wall is BC=1.6 meters, the distance from point D to the wall is DE=1.4 meters, and ##BD##=0.55 meters.

2. According to the problem statement, AB forms a right triangle △ABC with the ground and the vertical wall, and each rung of the ladder is parallel to the ground, which means DE is parallel to BC.

3. ##According to the theorem of proportional segments in parallel lines, since the line DE is parallel to the side BC of triangle ABC and intersects its other two sides AB and AC, the segments it intercepts are proportional to the corresponding segments of the original triangle, so AD/AB=DE/BC##.

####

##4##. Let the length of the ladder AB be x meters, then ##(x-0.55)/x = 1.4/1.6##.

##5##. Solving this proportion equation, we get: ##1.6(x-0.55)=1.4x, thus x =## 4.4##.

####

##6##. Finally, we find that the length of the ladder is 4.40 meters.

直角三角形; 线段; 余弦

Definition of Parallel Lines

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

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

Proportional Segments Theorem

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

In the triangle <note>ABC</note>, the <note>line DE is parallel to side BC</note> and intersects the other two sides AB and AC at <note>points D and E</note>. According to the Proportional Segments Theorem, we have: <note>AD/AB = DE/BC</note>, that is, the intercepted segments are proportional to the corresponding segments of the original triangle.


Question: m \angle 9 = 2 x - 4, m \angle 10 = 2 x + 4. 求 \angle 10 的度数。

Answer: 94

1. Given conditions are ∠9 = 2x - 4 and ∠10 = 2x + 4, let the vertical segment be AB and the horizontal segment be CD, with the two segments intersecting at point O.

2. According to the definition of adjacent supplementary angles, ∠9 and ∠10 are adjacent supplementary angles. Therefore, we can establish the equation (2x - 4) + (2x + 4) = 180.

3. Simplifying the equation, we get 4x = 180.

4. Thus, we get x = 45.

5. Substituting the value of x, 45, into the expression for ∠10, ∠10 = 2x + 4, we get ∠10 = 2(45) + 4 = 90 + 4 = 94.

6. Through the above reasoning, we finally conclude that the measure of ∠10 is 94 degrees.

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>In the figure of this problem, ∠9 and ∠10 share a common side</note>, <note>their other sides are extensions in opposite directions</note>, <note>so ∠9 and ∠10 are adjacent supplementary angles</note>. Therefore, according to the theorem, <note>∠9 + ∠10 = 180 degrees</note>. Specifically, (2x - 4) + (2x + 4) = 180 degrees.

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 diagram of this problem, <note>angle 9 is a geometric figure formed by two rays BO and CO</note>, which share a <note>common endpoint O</note>. This <note>common endpoint O is called the vertex of angle 9</note>, and <note>rays BO and CO are called the sides of angle 9</note>. Similarly, <note>angle 10 is a geometric figure formed by two rays AO and CO</note>, which share a <note>common endpoint O</note>. This <note>common endpoint O is called the vertex of angle 10</note>, and <note>rays AO and CO are called the sides of angle 10</note>.


Question: Find the area of the figure.

Answer: 120

1. From the figure, it can be seen that all four angles of the shape are 90 degrees. According to the definition of a rectangle, this shape is a rectangle. Additionally, the length of the rectangle in the figure is 12 ft and the width is 10 ft.

2. According to the area formula of a rectangle, which is length multiplied by width, the area of the rectangle in this problem is 12 ft multiplied by 10 ft.

3. The calculated area of the rectangle is 120 square feet.

4. Through the above reasoning, the final answer is 120 square feet.

Definition of Rectangle

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

In the figure of this problem, the quadrilateral is a <note>rectangle</note>, whose <note>internal angles are all right angles (90 degrees)</note>, and <note>two sides with a length of 12 feet are parallel and equal</note>, <note>two sides with a length of 10 feet are parallel and equal</note>.

Formula for the Area of a Rectangle

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

<note>The length of the rectangle is 12 feet, and the width is 10 feet</note>, so <note>the area of the rectangle</note> = <note>12 * 10 = 120 square feet</note>.


Question: Find PS.

Answer: 9

1. Given conditions: PR is the height of triangle PQS, PR is perpendicular to SQ, SQ=14, PQ=9, QR=7, SR=7.

2. ##Since PR is perpendicular to SQ, ∠PRQ=∠PRS=90°, according to the definition of a right triangle, triangles PRQ and PRS are both right triangles##.

3. ##Since RQ and SR are both 7, RQ=SR. Triangles PRQ and PRS share PR as the right angle side. According to the congruence theorem (SAS), since RQ=SR, PR=PR, ∠PRQ=∠PRS=90°, triangles PRQ and PRS are congruent##.

##4. According to the definition of congruent triangles, the corresponding sides of the two triangles are equal, so PS=PQ. Given PQ=9, PS=9##.

等腰三角形; 垂线; 直角三角形; 中点

Definition of Triangle

<note>Triangle PRS is a geometric figure composed of three non-collinear points P, R, S and their connecting line segments PR, PS, SR</note>。<note>Points P, R, S are the three vertices of the triangle</note>，<note>Line segments PR, PS, SR are the three sides of the triangle</note>。<note>Triangle PRQ is a geometric figure composed of three non-collinear points P, R, Q and their connecting line segments PR, PQ, QR</note>。<note>Points P, R, Q are the three vertices of the triangle</note>，<note>Line segments PR, PQ, QR are the three sides of the triangle</note>。

Definition of Right Triangle

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

In the figure of this problem, in triangle PQR, <note>angle PRQ is a right angle (90 degrees)</note>, so triangle PQR is a right triangle. <note>Sides PR and QR are the legs, and side PQ is the hypotenuse</note>. In triangle PSR, <note>angle PRS is a right angle (90 degrees)</note>, so triangle PSR is a right triangle. <note>Sides PR and SR are the legs, and side PS is the hypotenuse</note>.

Triangular Congruence Theorem (SAS)

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

<note>Side PR is equal to side PR</note>, <note>side SR is equal to side QR</note>, and <note>angle PRS is equal to angle PRQ</note>, therefore according to the Triangular Congruence Theorem (SAS), these two triangles are congruent.

Definition of Congruent Triangles

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

The original text: <note>Triangle PRQ and triangle PRS are congruent triangles</note>, the corresponding sides and corresponding angles of triangle PRQ are equal to those of triangle PRS, namely:
Side <note>PS = PQ</note>
Side <note>PR = PR</note>
Side <note>SR = QR</note>,
At the same time, the corresponding angles are also equal:
Angle <note>PRS = PRQ</note>
Angle <note>PSR = PQR</note>
Angle <note>SPR = QPR</note>.


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

Answer: 10.2

1. Given that this is a parallelogram, ##the figure shows a right angle mark within one angle of the parallelogram, indicating that these two lines are perpendicular to each other, i.e., the angle is a right angle. Additionally, there are short lines drawn on each of the four sides, indicating that these two sides are of equal length##.

2. ##According to the properties theorem of parallelograms and the supplementary angles theorem of parallelograms, the right angle and the opposite angle are equal, and the two adjacent angles are supplementary, so all four angles within the quadrilateral are right angles##.

3. ##By reverse definition of a rectangle, a rectangle is a type of quadrilateral with all four sides equal and all four internal angles being right angles (90 degrees). Since all four internal angles of the parallelogram are right angles, the parallelogram is also a rectangle##.

4. ##According to the area formula of a rectangle, the area of a rectangle = length * height. Given that the side length of the parallelogram is 3.2m and all four sides are equal, the area of the parallelogram = 3.2 * 3.2 = 10.24##. According to the specified requirement, the area is rounded to one decimal place, and 10.24 rounded off is 10.2 square meters.

5. Through the above reasoning, the final answer is 10.2 square meters.

Properties of Parallelogram Theorem

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

<note>In the figure of this problem, let the parallelogram be parallelogram ABCD</note>, the intersection point of the diagonals is <note>O</note>. In parallelogram ABCD, <note>diagonals AC and BD are equal</note>, <note>angles B and D are equal</note>; <note>sides AB and CD are equal</note>, <note>sides AD and BC are equal</note>; the diagonals <note>AC and BD bisect each other</note>, that is, the intersection point divides the diagonal <note>AC into two equal segments AO and CO</note>, <note>and divides the diagonal BD into two equal segments BO and DO</note>.

Definition of Parallelogram

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

In the figure of this problem, <note>quadrilateral ABCD is a parallelogram</note>, <note>side AB is parallel and equal to side CD</note>, <note>side AD is parallel and equal to side BC</note>.

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

In <note>parallelogram ABCD</note>, <note>angle A</note> and <note>angle D</note> are adjacent interior angles, <note>angle C</note> and <note>angle D</note> are also adjacent interior angles. According to the Adjacent Angles Supplementary Theorem of Parallelogram, <note>angle A + angle D = 180°</note>, <note>angle C + angle D = 180°</note>.

Definition of Rectangle

<note>Quadrilateral ABCD is a rectangle</note>, with its interior angles <note>∠A, ∠B, ∠C, ∠D all being right angles (90 degrees)</note>, and <note>sides AB and CD are parallel and equal in length</note>, <note>sides AD and BC are parallel and equal in length</note>.

Formula for the Area of a Rectangle

In the figure of this problem, <note>the area of a rectangle is equal to its length multiplied by its width</note>. Since <note>the figure is a rectangle</note> with each side measuring 3.2 meters, <note>area = length × width</note> = 3.2 meters × 3.2 meters = 10.24 square meters.


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

Answer: 41.8

1. According to the definition of a right triangle, △ABC is a right triangle, ∠ACB=90°, sides BC and AC are the legs, and side AB is the hypotenuse.

2. Given that side BC=18 and side AB=27. According to the definition of the sine function, the sine value of ∠CAB is the ratio of the opposite side to the hypotenuse, i.e., sin(∠CAB)= BC/ AB=18/27.

3. By calculation, sin(∠CAB)=18/27=2/3.

4. Using the inverse sine function (denoted as sin^{-1} or arcsin), the specific degree of ∠CAB can be obtained as sin^{-1}(2/3).

5. Calculating sin^{-1}(2/3), accurate to the tenths place, the approximate value of ∠CAB is 41.8°.

6. Through the above reasoning, the final answer is ∠CAB ≈ 41.8°, i.e., the degree of ∠A is approximately 41.8°.

直角三角形; 正弦; 余弦; 正切

Definition of Right Triangle

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

Definition of Sine Function

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

In the figure of this problem, <note>in the right triangle ABC, ∠CAB is an acute angle, side BC is the opposite side of ∠CAB, and side AB is the hypotenuse. According to the definition of the sine function, the sine value of ∠CAB is equal to the ratio of the opposite side BC to the hypotenuse AB</note>, i.e., sin(∠CAB) = BC / AB = 18 / 27.


Question: As shown in the figure, it is known that the inscribed angle ∠BAD=50°, then the degree of the inscribed angle ∠BCD is ()

Answer: 130°

1. From the figure, points A, B, C, and D are all on circle O. According to the inscribed quadrilateral, quadrilateral ABCD is inscribed in circle O.

2. According to the theorem (corollary 3 of the inscribed angle theorem) that the opposite angles of an inscribed quadrilateral are supplementary, ∠BAD and ∠BCD are supplementary, i.e., ∠BAD + ∠BCD = 180°.

3. Given ∠BAD = 50°, substitute into the equation from the previous step to get: 50° + ∠BCD = 180°.

4. Solve the equation to get: ∠BCD = 180° - 50° = 130°.

5. Therefore, the measure of the inscribed angle ∠BCD is 130°.

圆; 圆周角; 圆内接四边形

Cyclic Quadrilateral

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

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

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

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