Question:
Look at the diagram.

Which equation can be used to solve for x?

- 4x=76
- 4x+76=90
- 4x+76=180
- 4+x=76
Solve for x.

x= $\Box$

Answer:
- 4x=76
x=19

1. Observe the figure, it is known that ∠DHF and ∠EHG are vertical angles.

2. According to the definition of vertical angles, ∠DHF=∠EHG. Definition of vertical angles: When two lines intersect, the vertical angles formed are equal.

3. It is known from the problem that ∠DHF=76°, therefore ∠EHG=76°.

4. According to the information marked in the figure, ∠EHG=4x degrees.

5. We have the equation: 4x = 76.

6. Solving the equation for x, we get 4x = 76.

7. Using algebraic equation solving, first divide both sides by 4, we get x = 76 / 4.

8. Calculating 76/4, we get x = 19.

9. Through the above reasoning, the final answer is x = 19.

Definition of Vertical Angles

In the diagram of this problem, two intersecting lines <note>DG</note> and <note>EF</note> intersect at <note>point H</note>, forming four angles: <note>∠DHF</note>, <note>∠EHG</note>, <note>∠FHG</note>, and <note>∠EHD</note>. According to the definition of vertical angles, <note>∠DHF and ∠EHG are vertical angles</note>, <note>∠FHG and ∠EHD are vertical angles</note>. Since vertical angles are equal, <note>∠DHF=∠EHG</note>, <note>∠FHG=∠EHD</note>. Given that <note>∠DHF=76°</note>, therefore <note>∠EHG=76°</note>.


Question:
Look at this angle:

Name this angle:  $\Box$  $\angle $

Answer: \$\angle \$ TSR

1. In the geometric figure, we first identify the three points that form the angle: R, S, T.

2. In naming angles, the name of the angle usually consists of the three points that form the angle, with the vertex letter in the middle. According to this rule, we need to find the vertex of the angle.

3. Observing the figure, we can see that S is the common point of the two rays, so S is the vertex of the angle.

4. To fully name the angle, we need to determine the two sides of this angle, which are the segment SR from the vertex S to the left to point R, and the segment ST from the vertex S to the right to point T.

5. Therefore, the angle can be named ∠TSR or ∠RST, as long as S is in the middle.

6. According to the naming principles of angles in geometry, the angle can also be identified by just the vertex, so the angle can also be named ∠S.

7. Through the above reasoning, the answer is ∠TSR or ∠S.

Definition of an Angle

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

<note>Angle ∠TSR is a geometric figure composed of two rays SR and ST</note>, these two rays share a common endpoint <note>S</note>. This common endpoint <note>S is called the vertex of angle ∠TSR</note>, and the rays <note>SR and ST are called the sides of angle ∠TSR</note>.

Naming of Angles

An angle can be named using three points, with the vertex point located in the middle, or it can be named solely by the vertex.

<note>Angle ∠TSR</note> can be named as ∠S using <note>vertex S</note>, or named as ∠TSR using <note>the three points forming the angle</note>, with <note>S in the middle</note>.


Question: What is the measurement of this angle? $\Box$ °

Answer: 125°

1. First, identify the two scales of the protractor given in the figure. The inner scale is in the counterclockwise direction, and the outer scale is in the clockwise direction.

2. Determine the position of the baseline ray. In this figure, one ray coincides with the 0° line on the inner scale, and this ray also coincides with the 180° line on the outer scale.

3. Observe the intersection of the other ray with the inner scale to determine the angle measurement.

4. This ray passes through the 125° position on the inner scale.

5. Based on the angle measurement, the degree of ∠AOB is 125°.

Through the above reasoning, the final answer is 125°.

Definition of an Angle

The vertex <note>O</note> and the <note>two rays</note> starting from <note>vertex O</note> form an angle, where one ray coincides with the <note>0° line of the inner scale ring</note> and the other ray passes through the <note>125° position on the inner scale ring</note>. Therefore, the measure of this angle is <note>125°</note>.

Measurement of Angle

Align the baseline of the protractor with one side of the angle, then extend the other side to the protractor's scale. Determine the angle by reading the difference between the graduations of the two sides.

<note>The reference ray coincides with the 0° line of the inner scale ring</note>, <note>the other ray passes through the 125° position on the inner scale ring</note>, according to the protractor usage theorem, <note>the degree of angle ∠AOB is 125°</note>.


Question: What is the measurement of this angle? $\Box$ °

Answer: 135°

1. Given that one ray of the protractor aligns with 0° on the inner circle, this position corresponds to 180° on the outer circle.

2. Observe the reading of the other ray on the inner circle, which is the measure of the angle.

3. The other ray intersects at 135° on the inner circle.

4. Therefore, ∠ABC = 135°.

5. Through the above reasoning, the final answer is 135°.

Definition of an Angle

<note>An angle is a geometric figure formed by two rays</note>, these two rays <note>have a common endpoint, which is the center point of the protractor</note>. One end of the ray <note>aligns with 0° on the inner circumference</note>, and the other end <note>aligns with 135° on the inner circumference</note>, thus the formed angle is <note>135°</note>.

Definition of Ray

A ray is a part of a line that starts at a specific point and extends infinitely in one direction. It has a starting point but no endpoint.

Original text: <note>two rays</note>, where <note>one ray aligns with 0° on the inner circumference</note>, <note>the other ray aligns with 135° on the inner circumference</note>, these two rays form <note>angle ∠ABC</note>.

Measurement of Angle

<note>A ray aligns with 0° on the inner circumference</note>, <note>another ray aligns with 135° on the inner circumference</note>, therefore, the displayed angle is <note>135°</note>.


Question:
Look at this angle:

What is the vertex?
 $\Box$

Answer: X

1. According to the definition of an angle, an angle is formed by two rays that share a common endpoint, which is called the vertex of the angle.

2. In the figure provided in the problem, there are two rays XY and XW.

3. Observing that rays XY and XW intersect at point X, point X is the common endpoint of the two rays.

4. Therefore, according to the definition of an angle, point X is the vertex of angle YXW.

5. Through the above reasoning, the final answer is point X.

Definition of an Angle

The angle <note>YXW</note> is a geometric figure formed by two rays <note>XY and XW</note> that share a common endpoint <note>X</note>. This common endpoint <note>X</note> is called the vertex of the angle <note>YXW</note>, and the rays <note>XY and XW</note> are called the sides of the angle <note>YXW</note>.

Definition of Ray

<note>Ray XY starts from point X and extends infinitely in the direction of Y</note>, while <note>Ray XW also starts from point X and extends infinitely in the direction of W</note>.

Naming of Angles

<note>Angle YXW is represented by three letters</note>, where <note>letter X represents the vertex of the angle</note>, <note>letters Y and W represent the other endpoints of the two sides of the angle</note>, that is, <note>the endpoints of ray XY and ray XW</note>.


Question: What is the measurement of this angle? $\Box$ °

Answer: 100°

1. Observe the two rays on the protractor, one of which coincides with the 0° line on the inner circle of the protractor, which is the same as the 180° line on the outer circle, and this point is located on the right side of the protractor.

2. The other ray starts from the center of the protractor and points upwards to the position of 80° on the outer circle.

3. According to the measurement method of the protractor, the angle is measured on the inner circle, so it is necessary to read the angle where the other ray crosses the protractor on the inner circle.

4. Starting from 0° on the inner circle, look counterclockwise for the position where the second ray crosses the inner circle, and it can be seen that this ray crosses the inner circle at the 100° mark.

5. Therefore, the measure of the angle formed by the two rays is 100°.

6. After the above reasoning, the final answer is 100°.

Definition of an Angle

<note>An angle is a geometric figure formed by two rays, the 0° line and the 100° line</note>, <note>these two rays share a common endpoint</note>. This <note>common endpoint is called the vertex of the angle</note>, and <note>the rays 0° line and 100° line are called the sides of the angle</note>.

Measurement of Angle

<note>The center point of the protractor coincides with the vertex of the angle</note>, one side aligns with the <note>0° line of the inner circle of the protractor</note>, and the other side points to the <note>100° mark position of the inner circle</note>, indicating that the angle measures <note>100°</note>.


Question:
Look at this angle:

Name this angle:  $\Box$  $\angle $

Answer: \$\angle \$ IKJ

1. Observe the given angle and find that the vertex of the angle is at point K.

2. According to the definition of a geometric angle, an angle is formed by two rays that share a common endpoint, and this common endpoint is the vertex of the angle.

3. Confirm that the two rays forming the angle are the rays starting from point K passing through point I and the ray starting from point K passing through point J.

4. According to geometric naming rules, an angle can be named using three points, with the middle point being the vertex of the angle.

5. Therefore, the angle can be named ∠IKJ, where point K is the vertex.

6. According to another naming method, the angle can be named using only the vertex of the angle, thus another valid name is ∠K.

7. Through the above steps, it can be determined that the name of the angle is ∠IKJ or ∠K.

Definition of an Angle

In the figure of this problem, <note>angle ∠IKJ</note> is a geometric figure formed by two rays <note>KI</note> and <note>KJ</note>, which share a common endpoint <note>K</note>. This common endpoint <note>K</note> is called the <note>vertex of angle ∠IKJ</note>, and the rays <note>KI</note> and <note>KJ</note> are called the <note>sides of angle ∠IKJ</note>.

Naming of Angles

In the figure of this problem, according to the naming rules, the <note>vertex of the angle is point K</note>, and the angle can be named as <note>∠IKJ or ∠K</note>. <note>The middle point K is the vertex</note>, <note>points I and J are the endpoints of the angle</note>.

Definition of Ray

<note>Ray KI and ray KJ are two rays</note>, they <note>share a common starting point at point K</note>, and respectively <note>pass through points I and J</note>.


Question:
Look at this angle:

Name this angle:  $\Box$  $\angle $

Answer: \$\angle \$ SRQ

1. From the figure in the problem, it can be observed that the two rays of the angle are formed by connecting point S and point Q respectively to the common point R.

2. In geometry, the naming of an angle usually adopts the three-point naming method, with the vertex of the angle as the central letter. For example, in the given figure, the vertex is R.

3. Firstly, it can be named as angle S-R-Q, denoted in mathematical symbols as ∠SRQ, where S and Q are the extensions of the two rays of this angle at point R.

4. Secondly, the angle formed by these three points can be named from right to left, i.e., angle Q-R-S, denoted in mathematical symbols as ∠QRS.

5. Since the vertex is R, the letters at both ends can be ignored, and the angle can be named using only the vertex letter, i.e., angle R, denoted in mathematical symbols as ∠R.

6. In geometry, it is important to ensure that the middle letter always represents the vertex of the angle. Therefore, in this context, both ∠SRQ and ∠QRS correctly name the angle, avoiding ambiguity.

7. Through the above reasoning, it is confirmed that the name of the angle is ∠SRQ or ∠QRS, and it can also be abbreviated as ∠R.

Definition of an Angle

The angle <note>∠SRQ or ∠QRS</note> is a geometric figure formed by two <note>rays SR and QR</note>, which share a common endpoint <note>R</note>. This common <note>endpoint R</note> is called the vertex of the angle <note>∠SRQ or ∠QRS</note>, and the <note>rays SR and QR</note> are referred to as the sides of the angle <note>∠SRQ or ∠QRS</note>.

Naming of Angles

<note>The vertex is R, and the endpoints of the two rays are S and Q respectively</note>. Therefore, the angle can be named as <note>∠SRQ or ∠QRS</note>.


Question:
Look at this angle:

Name this angle:  $\Box$  $\angle $

Answer: \$\angle \$ TSU

1. Given that the vertex of the angle is at a point named S, and the angle is formed by two rays passing through points T, S, and U.

2. According to the principle that three points determine an angle, with the vertex being the middle point. Therefore, the angle can be represented as ∠TSU.

3. The name of the angle can also be uniquely determined by the vertex S, as it is the only explicitly stated vertex. Hence, the angle can also be represented as ∠S.

4. Through the above reasoning, the final answer is ∠TSU, which can also be represented as ∠S.

Definition of an Angle

<note>Angle ∠TSU is a geometric figure composed of two rays ST and SU</note>, these two rays have a common endpoint <note>S</note>. This common endpoint <note>S is called the vertex of angle ∠TSU</note>, and <note>rays ST and SU are called the sides of angle ∠TSU</note>.

Naming of Angles

In the diagram of this problem, <note>the vertex of the angle is point S</note>, <note>the two rays are ST and SU respectively</note>, so the name of the angle can be defined as <note>∠TSU</note>. Since <note>vertex S is unique</note>, this angle can also be simply referred to as <note>∠S</note>.


Question:
Look at this angle:

Name this angle:  $\Box$  $\angle $

Answer: \$\angle \$ WXY

1. The angle shown in the figure is formed at point X, which is the vertex of this angle.

2. To determine the name of the angle, three points are needed, with the vertex being in the middle. The two sides of this angle pass through point W and point Y respectively.

3. According to the definition of an angle, the angle can be named using these three points: ∠WXY, ensuring the vertex X is in the middle.

4. The same angle can also be named as: ∠YXW, still keeping the vertex X in the middle.

5. The angle can also be represented using only the vertex. In this case, it can be simplified to ∠X.

6. In summary, the angle can be named as: ∠WXY, ∠YXW, or ∠X.

7. Based on the above reasoning, the final answer is multiple names representing the angle: ∠WXY, ∠YXW, ∠X.

Definition of an Angle

<note>Angle WXY is a geometric figure formed by two rays WX and XY</note>, <note>these two rays share a common endpoint X</note>. <note>This common endpoint X is called the vertex of angle WXY</note>, and <note>rays WX and XY are called the sides of angle WXY</note>.

Naming of Angles

<note>Vertex X</note> and points on the sides <note>W and Y</note> can be used to name the angle as <note>angle WXY or angle YXW</note>, ensuring that <note>vertex X is in the middle</note>; it can also be simplified by using only the vertex, i.e., <note>angle X</note>.


Question:
Look at the diagram.

Which equation can be used to solve for x?

- 3(5+x)+60=180
- 3(5+x)=60
- 3(5x)+60=90
- 3(5x)=60
Solve for x.

x= $\Box$

Answer:
- 3(5+x)=60
x=15

1. From the figure, it can be seen that line AD and line FC intersect at point G, forming vertical angles ∠AGF and ∠CGD.

2. According to the definition of vertical angles, therefore ∠AGF = ∠CGD.

3. It is known that the degree of ∠AGF is 60 degrees, while the degree of ∠CGD is 3(5+x).

4. Since ∠AGF = ∠CGD = 60 degrees, so 3(5+x) = 60.

5. Solving the equation 3(5+x) = 60, first divide by 3: 5+x = 20.

6. Then subtract 5 from both sides of the equation to get x = 15.

7. Through the above reasoning, the final answer is x=15.

Definition of Vertical Angles

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

<note>Formed by the intersection of line AD and line FC at point G</note>, according to the vertical angles theorem, we can conclude that <note>vertical angle ∠AGF is equal to vertical angle ∠CGD</note>. Therefore, <note>∠AGF = ∠CGD</note>. In this problem, it is known that <note>∠AGF = 60 degrees</note>, and <note>∠CGD = 3(5+x)</note>.


Question: What is the measurement of this angle? $\Box$ °

Answer: 90°

1. First, according to the protractor's scale, it can be seen that the left scale starts at 0° and increases in the clockwise direction; the right scale starts at 180° and decreases in the counterclockwise direction.

2. Observing the two rays in the figure, it is found that one ray coincides with the 0° scale line of the protractor, i.e., this ray serves as the initial side of the angle.

3. Observing the intersection of the other ray with the protractor's scale, it can be seen that this ray passes through the 0° to 180° marking line of the protractor. At this point, in the inner scale, this ray points to the 90° position.

4. According to the definition of a right angle: two perpendicular rays form a 90° angle.

5. Further combining the judgment from the figure, it is confirmed that the other ray indeed passes through the 90° mark of the inner scale, so the angle measure is 90°.

6. Through the above reasoning, the final answer is 90°.

Definition of an Angle

In the figure of this problem, an angle is a geometric shape formed by two rays, <note>these two rays share a common endpoint</note>. This <note>common endpoint is the center point of the protractor</note>, called the <note>vertex of the angle</note>, and the rays are respectively <note>a ray coinciding with the 0° mark of the protractor</note> and <note>another ray pointing to the 90° position</note>.


Question:
Look at this angle:

What is the vertex?
 $\Box$

Answer: U

1. Observe the figure given in the problem, there is an angle in the figure.

2. According to the geometric definition, an angle is a figure formed by two rays starting from a common endpoint.

3. In the figure, ray UT and ray UV extend from point U to T and from point U to V respectively.

4. Therefore, the common endpoint of the two rays UT and UV is point U.

5. According to the definition of an angle, the vertex of the angle is the common endpoint of the two rays.

6. Hence, the vertex of the angle is point U.

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

<note>An angle ∠TUV is a geometric figure formed by two rays UT and UV</note>, these two rays <note>share a common endpoint U</note>. This common endpoint <note>U is called the vertex of angle ∠TUV</note>, and the rays <note>UT and UV are called the sides of angle ∠TUV</note>.


Question:
Look at this angle:

Name this angle:  $\Box$  $\angle $

Answer: \$\angle \$ ABC

1. According to the figure given in the problem, we can see an angle formed by three points A, B, and C.

2. In geometry, the name of an angle is usually represented starting with the vertex point, which is the point between the two lines forming the angle. Therefore, the vertex of the angle is point B.

3. The naming rule for angles is: start with the symbol ∠, then write the point on one side of the angle, the vertex, and the point on the other side of the angle in sequence.

4. According to the figure given in the problem, points A and C are connected to vertex B, so we can represent this angle in order as ∠ABC.

5. We can also write this angle as ∠CBA according to the naming rule, which represents the same angle.

6. Finally, since point B is the vertex of this angle, it can be abbreviated as ∠B, but to avoid ambiguity, three letters are usually used.

7. After the above analysis and description, we can conclude that the name of this angle is ∠ABC or ∠CBA.

The angle <note>∠ABC</note> is a geometric figure formed by two rays <note>AB and BC</note>, which share a <note>common endpoint B</note>. This common <note>endpoint B</note> is called the <note>vertex of angle ∠ABC</note>, and the rays <note>AB and BC</note> are referred to as the <note>sides of angle ∠ABC</note>.

Naming of Angles

In the diagram of this problem, this angle can be named as <note>∠ABC</note> or <note>∠CBA</note>. Because it is an angle formed by <note>point A and point C</note> through <note>vertex B</note>, it is represented in sequence as <note>∠ABC</note>. The vertex is <note>point B</note>, and the sides are <note>line segment AB</note> and <note>line segment BC</note>.


Question:
Find the measure of  $\angle $ ILJ.

Write your answer as a whole number or a decimal.

m $\angle $ ILJ= $\Box$ °

Answer: m \$\angle \$ ILJ=102°

1. Given that chord HI and chord JK intersect at point L, find the measure of ∠ILJ.

2. According to the theorem of angles inside a circle, we deduce that m∠ILK = 1/2(arc JH + arc IK).

3. Substituting the given arc measures, we get: m∠ILK = 1/2(67° + 89°) = 78°.

4. Notice that ∠ILJ and ∠ILK lie on a straight line, and ∠JLK is a straight angle, hence the sum of ∠ILJ and ∠ILK is 180°.

5. Therefore, m∠ILJ = 180° - m∠ILK = 180° - 78°.

6. Calculating this, we get m∠ILJ = 102°.

7. Through the above reasoning, the final answer is 102°.

Inscribed Angle Theorem

The measure of an inscribed angle in a circle is equal to half the sum of the measures of the arcs that the angle intercepts.

<note>Angle ILK</note> is the angle formed by two chords <note>JK and HI intersecting inside the circle</note>, so <note>Angle ILK is an inscribed angle</note>, that is, <note>Angle ILK measure = 1/2(arc JH + arc IK)</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.

Original text: <note>Ray LJ and ray JK are on the same straight line</note>, <note>angle JLK is a straight angle</note>, with a measure of <note>180 degrees</note>.


Question:
Look at this angle:

Name this angle:  $\Box$  $\angle $

Answer: \$\angle \$ ACB

1. Observe the figure, noting that there are three points: A, B, and C, and they are not collinear.

2. According to the definition of a geometric angle, two rays form an angle when they have a common endpoint. Here, the common endpoint of rays CA and CB is point C.

3. To name this angle, we use three points to describe it, with the common endpoint (vertex) in the middle. Therefore, the angle can be denoted as ∠ACB or ∠BCA.

4. Another naming method is to use only the vertex to name the angle, especially when there is no ambiguity. Thus, the angle can also be simply named as ∠C.

5. After the above analysis, this angle can be named as ∠ACB, ∠BCA, or ∠C.

Definition of an Angle

<note>Angle ∠ACB is a geometric figure formed by two rays CA and CB</note>, which share a common endpoint <note>C</note>. This common <note>endpoint C is called the vertex of angle ∠ACB</note>, and <note>rays CA and CB are called the sides of angle ∠ACB</note>.

Naming of Angles

<note>The vertex of the angle is point C</note>, and the three points involved are <note>point A, point B, and point C</note>. Therefore, the angle can be named as <note>∠ACB or ∠BCA</note>. Additionally, when there is no ambiguity, the angle can also be named as <note>∠C</note>.


Question:
△RST and △UVW are shown below.

Which statement is true?

- △RST is similar to △UVW.
- △RST is not similar to △UVW.
- There is not enough information to determine whether the triangles are similar.

Answer: - △RST is similar to △UVW.

1. Given in triangle RST, ∠S = 36°, ∠T = 72°, find the angle of ∠R. According to the triangle angle sum theorem, the sum of the three interior angles of a triangle is 180°.

2. Using the triangle angle sum theorem, let ∠R be x, then x + 36° + 72° = 180°. Therefore, we get x = 180° - 108° = 72°, i.e., ∠R = 72°.

3. After calculation, the three angles of △RST are ∠R = 72°, ∠S = 36°, ∠T = 72°.

4. Given in △UVW, ∠V = 72°, ∠W = 72°, according to the triangle angle sum theorem, the third angle ∠U = 180° - 72° - 72° = 36°.

5. △RST and △UVW each have two pairs of corresponding angles equal, which are ∠R = ∠U = 72°, ∠T = ∠W = 72°, ∠S = ∠V = 36°.

6. According to the similarity criterion theorem (AA), when two corresponding interior angles of two triangles are equal, these two triangles are similar triangles, therefore △RST and △UVW are similar.

7. Through the above reasoning, the final answer is △RST is similar to △UVW.

Triangle Angle Sum Theorem

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

In the diagram of this problem, in <note>triangle RST</note>, <note>angle S, angle T, and angle R</note> are the three interior angles of <note>triangle RST</note>. According to the Triangle Angle Sum Theorem, <note>angle S + angle T + angle R = 180°</note>. Similarly, in <note>triangle UVW</note>, <note>angle U, angle V, and angle W</note> are the three interior angles of <note>triangle UVW</note>. According to the Triangle Angle Sum Theorem, <note>angle U + angle V + angle W = 180°</note>.

Similarity Theorem for Triangles (AA)

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

<note>△RST and △UVW have two corresponding angles equal:</note> ∠R = ∠U = 72°, <note>∠T = ∠W = 72°</note>, and <note>the third angle is also equal</note>, ∠S = ∠V = 36°. Therefore, according to the <note>AA theorem</note>, <note>△RST and △UVW are similar triangles</note>.


Question:
△BCD and △FED are shown below.

Which statement is true?

- △BCD is similar to △FED.
- △BCD is not similar to △FED.
- There is not enough information to determine whether the triangles are similar.

Answer: - △BCD is similar to △FED.

1. Given △BCD, ∠B measures 49°, and ∠C is a right angle, i.e., 90°.

2. According to the triangle angle sum theorem, which states that the sum of the three interior angles of a triangle is 180°, we can deduce: ∠B + ∠C + ∠BDC = 180°.

3. Substituting the known angles, i.e., 49° + 90° + ∠BDC = 180°, we can solve for ∠BDC = 41°.

4. Therefore, the three interior angles of △BCD are 49°, 90°, and 41°.

5. Observing △FED, it is known that ∠FED = 90° and ∠EDF = 41°.

6. Comparing the angles of △FED and △BCD, we can find that they have two identical angles: ∠FED = ∠C and ∠EDF = ∠BDC.

7. According to the similarity criterion for triangles (AA), if two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar. This definition is commonly referred to as the

Triangle Angle Sum Theorem

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

Similarity Theorem for Triangles (AA)

In the diagram of this problem, in the diagram of this problem, in triangles BCD and DEF, if angle BDC is equal to angle FDE, and angle BCD is equal to angle FED, then triangle BCD is similar to triangle DEF.


Question:
Find the measure of  $\angle $ MPO.

Write your answer as a whole number or a decimal.

m $\angle $ MPO= $\Box$ °

Answer: m \$\angle \$ MPO=116°

1. Given that the measure of arc LO is 96°, the measure of arc NM is 32°, and chords LM and NO intersect at point P.

2. According to the inscribed angle theorem, the measure of angle ∠LPO is equal to half the sum of the measures of the arcs intercepted by the angle, i.e., ∠LPO = 1/2(arc LO + arc NM).

3. Substituting the given arc measures into the formula, we get ∠LPO = 1/2(96° + 32°) = 1/2(128°) = 64°.

4. Since ∠LPO and ∠MPO are adjacent supplementary angles on the straight line NO.

6. Substituting the value of ∠LPO, we get ∠MPO = 180° - 64° = 116°.

7. Through the above reasoning, we finally obtain that the measure of ∠MPO is 116°.

Inscribed Angle Theorem

<note>Angle LPO</note> is the angle formed by <note>two chords LM and N intersecting within the circle</note>, so <note>angle LPO is an inscribed angle</note>, that is, <note>angle LPO = 1/2 (arc LO + arc NM)</note>

Definition of Linear Pair of Angles

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

In the figure of this problem, <note>angle LPO and angle MPO share a common side PO</note>, <note>their other sides LP and PM are extensions in opposite directions</note>, so angle LPO and MPO are adjacent supplementary angles.


Question:
△ABC and △DEF are shown below.

Which statement is true?

- △ABC is similar to △DEF.
- △ABC is not similar to △DEF.
- There is not enough information to determine whether the triangles are similar.

Answer: - △ABC is similar to △DEF.

1. In △ABC, it is known that ∠BAC=15° and ∠ACB=45°. According to the triangle angle sum theorem, the sum of the three interior angles of the triangle is 180°.

2. According to the triangle angle sum theorem, let ∠ABC be x, then: ∠BAC + ∠ABC + ∠ACB = 180°, that is, 15° + x + 45° = 180°.

3. Calculating, we get x = 180° - 15° - 45° = 120°, therefore ∠ABC=120°.

4. In △DEF, it is known that ∠EFD=45° and ∠FED=120°.

5. Comparing the corresponding angles of the two triangles, ∠ABC corresponds to ∠FED and they are equal (120°), ∠ACB corresponds to ∠EFD and they are equal (45°).

6. According to the AA similarity theorem, if two triangles have two pairs of corresponding angles that are equal, then the two triangles are similar. In this problem, ∠ABC and ∠FED are equal, ∠ACB and ∠EFD are equal, therefore △ABC and △DEF are similar.

7. From the above reasoning, we can conclude that △ABC and △DEF are similar.

Triangle Angle Sum Theorem

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

Similarity Theorem for Triangles (AA)

In the figure of this problem, by verifying the corresponding angles of △ABC and △DEF: <note>∠ABC and ∠DEF are equal</note>, <note>∠ACB and ∠DFE are equal</note>. Therefore, according to the similarity theorem for triangles, these two triangles are similar, the conclusion is <note>△ABC and △DEF are similar</note>.


Question:
Find the measure of minor arc $\overset{\frown}{SW}$ .

Write your answer as a whole number or a decimal.

m $\overset{\frown}{SW}$ = $\Box$ °

Answer: m \$\overset{\frown}{SW}\$ =137°

1. The figure shows the secants US and UW intersecting at point U.

2. It is known that angle ∠SUW = 42°, arc m( ⌒ TV ) = 53°.

3. According to the external angle theorem of a circle, the angle formed by the secants is equal to half the difference of the intercepted arcs. The formula for this theorem is: m(∠SUW) = 1/2 [m( ⌒ SW ) - m( ⌒ TV )].

4. Substitute the known values: 42° = 1/2 [m( ⌒ SW ) - 53°].

5. Solve the equation: multiply both sides of the equation by 2: 84° = m( ⌒ SW ) - 53°.

6. Continue solving the equation: 84° + 53° = m( ⌒ SW ).

7. Calculate to get: m( ⌒ SW ) = 137°.

8. Through the above reasoning, the final answer is 137°.

Definition of Arc

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

In the figure of this problem, there are two points <note>T and V</note> on the circle, and <note>arc TV</note> is a segment of the curve connecting these two points. According to the definition of an arc, <note>arc TV</note> is a segment of the curve between the two points <note>T and V</note> on the circle.

External Angle Theorem of a Circle

The measure of an exterior angle of a circle is equal to one-half the difference of the measures of the intercepted arcs.

<note>Angle SUW is an external angle of the circle</note>, <note>arc TV and arc SW are the two arcs opposite to angle SUW</note>. According to the External Angle Theorem of a Circle, angle SUW is equal to half the difference between the degrees of arc TV and arc SW, that is, <note>angle SUW = (degrees of arc SW - degrees of arc TV) / 2</note>.


Question:
IJ is tangent to the circle at J. Find the measure of  $\angle $ HIJ.

Write your answer as a whole number or a decimal.

m $\angle $ HIJ= $\Box$ °

Answer: m \$\angle \$ HIJ=44.5°

1. Given that IJ is a tangent to the circle and touches it at point J, according to the problem statement, the three arcs on the circle are \\overset{\\frown}{GH} (93°), \\overset{\\frown}{GJ} (178°), and \\overset{\\frown}{HJ}. By adding them, we get the total degrees of the circle as 360°.

2. According to the properties of the circle's angles, we know 93° + 178° + m \\overset{\\frown}{HJ} = 360°.

3. By calculation, we get m \\overset{\\frown}{HJ} = 360° - 93° - 178° = 89°.

4. Given that the tangent IJ intersects the chord IG at I, according to the external angle theorem of the circle, the angle between the tangent and the chord (m \\angle HIJ) is equal to half the difference between the arc intercepted by the chord (m \\overset{\\frown}{GJ}) and the corresponding arc opposite the chord (m \\overset{\\frown}{HJ}).

5. By the external angle theorem of the circle, m \\angle HIJ = \\frac{1}{2}(m \\overset{\\frown}{GJ} - m \\overset{\\frown}{HJ}).

6. Substituting the known degrees of the arcs, we get m \\angle HIJ = \\frac{1}{2}(178° - 89°).

7. Calculating, we get m \\angle HIJ = \\frac{1}{2}(89°) = 44.5°.

8. Through the above reasoning, we finally get the answer as 44.5°.

Angle Properties of a Circle

The total measure of the angles in a complete circular arc is 360 degrees, and the sum of the central angles subtended by the entire circumference of a circle is 360 degrees.

<note>Arc GH, arc HJ, arc GJ around the circle</note>, the sum of the radians is <note>360 degrees</note>, and the sum of the corresponding central angles is also <note>360 degrees</note>. Let the center of the circle be <note>O</note>, that is, <note>angle GOH + angle HOJ + angle GOJ = 360 degrees</note>, then <note>angle GOH = 93 degrees</note>.

External Angle Theorem of a Circle

<note>Let the center of the circle be O</note>, in circle O, <note>angle HIJ is the external angle of the circle</note>, <note>arc HJ and arc GJ are the two arcs opposite to angle HIJ</note>. According to the External Angle Theorem of a Circle, <note>angle HIJ is equal to half the difference between the degree measures of arc GJ and arc HJ</note>, that is, <note>angle HIJ = (degree measure of arc GJ - degree measure of arc HJ) / 2</note>.


Question:
△JKL and △MNO are shown below.

Which statement is true?

- △JKL is similar to △MNO.
- △JKL is not similar to △MNO.
- There is not enough information to determine whether the triangles are similar.

Answer: - △JKL is not similar to △MNO.

1. According to the information in the problem image, it is known that in △JKL, ∠K = 27°, ∠L = 37°.

2. Using the triangle angle sum theorem, which states that the sum of the three interior angles of a triangle is 180°, find the degree of ∠J. The specific steps are as follows:

According to the triangle angle sum theorem, we have: ∠J + ∠K + ∠L = 180°.

Substitute the known angles into the equation: ∠J + 27° + 37° = 180°.

Subtract 64° from both sides to get: ∠J = 116°.

3. Therefore, the angles of triangle △JKL are: ∠J = 116°, ∠K = 27°, ∠L = 37°.

4. Referring to the problem image, in △MNO, the interior angles are: ∠M = 126°, ∠N = 27°, and according to the triangle angle sum theorem, ∠O = 180° - ∠M - ∠N = 180° - 126° - 27° = 27°.

5. The angles of triangle △MNO are: ∠M = 126°, ∠N = 27°, ∠O = 27°.

6. By comparing the angles, it can be found that △JKL and △MNO do not have the same set of interior angles, because △JKL does not have a 126° angle, and △MNO does not have a 116° angle.

7. According to the criteria for determining similar triangles, in this problem, using the angle-angle (AA) similarity theorem, which states that if two triangles have corresponding angles that are equal, then the triangles are similar, cannot be satisfied.

8. After the above reasoning, the final answer is △JKL is not similar to △MNO.

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 diagram of this problem, the triangle <note>△JKL</note> is a geometric figure composed of three non-collinear points <note>J</note>, <note>K</note>, <note>L</note> and their connecting line segments <note>JK</note>, <note>KL</note>, <note>LJ</note>. Points <note>J</note>, <note>K</note>, <note>L</note> are the three vertices of the triangle, and line segments <note>JK</note>, <note>KL</note>, <note>LJ</note> are the three sides of the triangle. The triangle <note>△MNO</note> is a geometric figure composed of three non-collinear points <note>M</note>, <note>N</note>, <note>O</note> and their connecting line segments <note>MN</note>, <note>NO</note>, <note>OM</note>. Points <note>M</note>, <note>N</note>, <note>O</note> are the three vertices of the triangle, and line segments <note>MN</note>, <note>NO</note>, <note>OM</note> are the three sides of the triangle.

Triangle Angle Sum Theorem

In the figure of this problem, in triangle <note>△JKL</note>, <note>angle J</note>, <note>angle K</note>, and <note>angle L</note> are the three interior angles of triangle <note>△JKL</note>. According to the Triangle Angle Sum Theorem, <note>angle J + angle K + angle L = 180°</note>. Specifically calculated as <note>∠J + 27° + 37° = 180°</note>, so <note>∠J = 116°</note>. Correspondingly, in triangle <note>△MNO</note>, <note>angle M</note>, <note>angle N</note>, and <note>angle O</note> are the three interior angles of triangle <note>△MNO</note>. According to the Triangle Angle Sum Theorem, <note>angle M + angle N + angle O = 180°</note>. Specifically calculated as <note>126° + 27° + ∠O = 180°</note>, so <note>∠O = 27°</note>.

Similarity Theorem for Triangles (AA)

<note>Angle K equals angle N = 27°</note>, so triangle JKL and triangle MNO are not similar.


Question:
What is m $\angle $ H?

m $\angle $ H= $\Box$ °

Answer: m \$\angle \$ H=73°

1. Given quadrilateral EFGH is a cyclic quadrilateral, according to the theorem (Corollary 3 of the Inscribed Angle Theorem) that the opposite angles of a cyclic quadrilateral are supplementary, the opposite angles of a cyclic quadrilateral are supplementary, i.e., ∠EFG + ∠EHG = 180°.

2. Given ∠EFG = 107°, substitute it into the supplementary angle formula to get 107° + ∠EHG = 180°.

3. From step 2, we can calculate ∠EHG = 180° - 107°.

4. The calculation result is ∠EHG = 73°.

5. Through the above reasoning, the final answer is ∠EHG = 73°.

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

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

In the <note>cyclic quadrilateral EFGH</note>, the vertices E, F, G, and H are all on the circle. According to the <note>Diagonal Supplementary Theorem for Cyclic Quadrilateral</note>, the sum of each pair of opposite angles of the <note>quadrilateral EFGH</note> is equal to 180°. Specifically, <note>∠EFG + ∠EHG = 180°</note>; <note>∠FEH + ∠FGH = 180°</note>.


Question:
△RST and △TUV are shown below.

Which statement is true?

- △RST is similar to △TUV.
- △RST is not similar to △TUV.
- There is not enough information to determine whether the triangles are similar.

Answer: - △RST is not similar to △TUV.

1. Given △RST, ∠R equals 54°, ∠S is a right angle and equals 90°.

2. According to the triangle angle sum theorem, the sum of the three interior angles of a triangle is 180°. Thus, we can find the measure of ∠RTS. Let ∠RTS be x degrees, then 54° + 90° + x = 180°.

3. Solving the equation, we get x = 36°. Therefore, ∠RTS equals 36°.

4. In △TUV, given ∠TVU equals 46°, ∠U is a right angle and equals 90°.

5. According to the similarity criterion for triangles (AA), if two triangles have corresponding angles equal, they are similar.

6. Compare the interior angles of △RST and △TUV:

7. If ∠R in △RST equals 54°, ∠S equals 90°, and ∠RTS equals 36°.

8. ∠TVU in △TUV equals 46°, which is clearly different from any angle in △RST. Therefore, we cannot confirm the similarity of △RST and △TUV based on the similarity conditions.

9. Based on the above reasoning, the final conclusion is that △RST is not similar to △TUV.

Definition of Triangle

In the diagram of this problem, the triangle <note>RST is a geometric figure composed of three non-collinear points R, S, T and their connecting line segments RS, ST, RT</note>. Points <note>R, S, T are respectively the three vertices of the triangle</note>, and line segments <note>RS, ST, RT are respectively the three sides of the triangle</note>. The triangle <note>TUV is a geometric figure composed of three non-collinear points T, U, V and their connecting line segments TU, UV, VT</note>. Points <note>T, U, V are respectively the three vertices of the triangle</note>, and line segments <note>TU, UV, VT are respectively the three sides of the triangle</note>.

Similarity Theorem for Triangles (AA)

<note>△RST and △TUV</note> have two <note>corresponding angles equal</note>, so they are similar. In this problem, we compare <note>∠RST and ∠TUV</note>, <note>∠SRT and ∠UVT</note>, <note>∠RTS and ∠VTU</note>. Since <note>∠UVT (46°)</note> is different from <note>∠SRT (54°)</note>, it is not possible to confirm the similarity of <note>△RST and △TUV</note> based on the similarity conditions for triangles.

Definition of Right Triangle

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

<note>In the diagram of this problem, in triangle RST, ∠RST is a right angle (90 degrees)</note>, therefore triangle RST is a right triangle. <note>Side RS and side ST are the legs</note>, <note>side RT is the hypotenuse</note>. In triangle TUV, <note>∠TUV is a right angle (90 degrees)</note>, therefore triangle TUV is a right triangle. <note>Side TU and side UV are the legs</note>, <note>side TV is the hypotenuse</note>.

Triangle Angle Sum Theorem

In <note>triangle RST</note>, <note>angle R (∠SRT)</note>, <note>angle S (∠RST)</note>, and <note>angle RTS</note> are the three interior angles of triangle RST. According to the <note>Triangle Angle Sum Theorem</note>, <note>angle R + angle S + angle RTS = 180°</note>.


Question:
△CDE and △FGH are shown below.

Which statement is true?

- △CDE is similar to △FGH.
- △CDE is not similar to △FGH.
- There is not enough information to determine whether the triangles are similar.

Answer: - △CDE is not similar to △FGH.

1. In △CDE, it is known that ∠C=48°, ∠E=80°.

2. According to the triangle angle sum theorem, that is, the sum of the three interior angles of a triangle is 180°, we have ∠C+∠D+∠E=180°.

3. Substitute the known angles, we get 48°+∠D+80°=180°.

4. Solve the above equation, ∠D=180°-48°-80°=52°.

5. Therefore, the three angles of triangle △CDE are 48°, 80°, 52°.

6. Observe the angles of △FGH, it is known that ∠F=48°, ∠G=62°.

7. Using a similar method, calculate ∠H using the triangle angle sum theorem: ∠F+∠G+∠H=180°.

8. Substitute the known angles, we get 48°+62°+∠H=180°.

9. Solve the above equation, ∠H=180°-48°-62°=70°.

10. Therefore, the three angles of triangle △FGH are 48°, 62°, 70°.

11. To determine if two triangles are similar, the corresponding angles must be equal. According to the above results, the interior angles of triangles △CDE and △FGH do not all correspond.

12. Observe the angles of triangle △CDE: 48°, 80°, 52° and △FGH: 48°, 62°, 70°.

13. Since the interior angles of the two triangles are not the same, according to the similarity triangle theorem (AA): if two triangles have two pairs of corresponding angles equal, then the two triangles are similar, it can be concluded that △CDE is not similar to △FGH.

14. After the above reasoning, the final answer is "△CDE is not similar to △FGH.".

Definition of Triangle

In the diagram of this problem, the triangle <note>△CDE</note> is a geometric figure composed of three non-collinear points <note>C, D, E</note> and their connecting line segments <note>CD, DE, EC</note>. Points <note>C, D, E</note> are the three vertices of the triangle, and line segments <note>CD, DE, EC</note> are the three sides of the triangle. The triangle <note>△FGH</note> is a geometric figure composed of three non-collinear points <note>F, G, H</note> and their connecting line segments <note>FG, GH, HF</note>. Points <note>F, G, H</note> are the three vertices of the triangle, and line segments <note>FG, GH, HF</note> are the three sides of the triangle.

Triangle Angle Sum Theorem

In the figure of this problem, in triangle <note>△CDE</note>, <note>angle C</note>, <note>angle D</note>, and <note>angle E</note> are the three interior angles of triangle <note>△CDE</note>. According to the Triangle Angle Sum Theorem, <note>angle C + angle D + angle E = 180°</note>. Substituting the known angles, we get <note>48° + ∠D + 80° = 180°</note>, thus solving for <note>∠D = 52°</note>; for triangle <note>△FGH</note>, we have <note>∠F + ∠G + ∠H = 180°</note>. Substituting the known angles, we get <note>48° + 62° + ∠H = 180°</note>, thus solving for <note>∠H = 70°</note>.

Similarity Theorem for Triangles (AA)

<note>The angles of triangle △CDE are 48°, 80°, and 52°</note>; <note>The angles of triangle △FGH are 48°, 62°, and 70°</note>. Comparing the interior angles of the two triangles, since the interior angles of these two triangles do not all correspond equally, according to the Similarity Theorem for Triangles (AA), <note>△CDE is not similar to △FGH</note>.


Question:
Find the measure of minor arc $\overset{\frown}{JN}$ .

Write your answer as a whole number or a decimal.

m $\overset{\frown}{JN}$ = $\Box$ °

Answer: m \$\overset{\frown}{JN}\$ =129°

1. Given LJ and LN are secants of the circle, point L is outside the circle, and their intersection points with the circle are J and N respectively.

2. According to the external angle theorem of the circle, we know that the measure of ∠JLN is equal to half the difference of the measures of the two arcs it intercepts (i.e., arc JN and arc KM).

3. Given ∠JLN=41°, ∠JLN=1/2(measure of arc JN − measure of arc KM).

4. Substitute the given conditions into the formula: 41°=1/2(measure of arc JN − 47°).

5. Solve the equation by first multiplying both sides by 2, obtaining 82°=measure of arc JN − 47°.

6. Move 47° to the other side and add it to both sides of the equation, obtaining measure of arc JN = 82° + 47°.

7. After calculation, the measure of arc JN = 129°.

8. Through the above reasoning, the final answer is 129°.

External Angle Theorem of a Circle

<note>∠JLN is the external angle of the circle</note>, <note>arc KM and arc JN are the two arcs subtended by ∠JLN</note>. According to the External Angle Theorem of a Circle, <note>∠JLN is equal to half the difference between the degree measures of arc JN and arc KM</note>, that is, <note>∠JLN=1/2 (degree measure of arc JN − degree measure of arc KM)</note>.

Definition of Arc

<note>There are two points J and N on the circle</note>, <note>Arc JN is a segment of the curve connecting these two points</note>. According to the definition of an arc, <note>Arc JN is a segment of the curve between the two points J and N on the circle</note>.

Definition of an Angle

<note>Angle JLN is a geometric figure formed by two rays LJ and LN</note>, these two rays share a common endpoint <note>L</note>. This common endpoint <note>L is called the vertex of angle JLN</note>, and <note>rays LJ and LN are called the sides of angle JLN</note>.


Question:
Find the measure of  $\angle $ NQO.

Write your answer as a whole number or a decimal.

m $\angle $ NQO= $\Box$ °

Answer: m \$\angle \$ NQO=102°

1. Given that chord MN and chord OP intersect at point Q, and the lengths of two arcs are provided: arc MO = 62°, arc NP = 94°.

2. According to the Circle Interior Angle Theorem, the degree of an interior angle of a circle is equal to half the sum of the degrees of the two arcs subtended by the angle. Therefore, we can calculate angle MQO. Thus, ∠MQO = 1/2(arc MO + arc NP).

3. Substitute the arc degrees into the formula to get ∠MQO = 1/2(62° + 94°) = 1/2(156°).

4. Calculate 1/2 * 156° to determine ∠MQO = 78°.

5. Since ∠MQO and ∠NQO are a pair of adjacent supplementary angles on a straight line, meaning the sum of two adjacent angles is 180 degrees. Therefore, according to the definition of supplementary angles, we can write: ∠NQO = 180° - ∠MQO.

6. Substitute the known ∠MQO = 78° into the equation to get ∠NQO = 180° - 78° = 102°.

7. Through the above reasoning, the final answer is 102°.

Definition of Arc

In the figure of this problem, there are two points <note>M and O</note> on the circle, and <note>arc MO</note> is a segment of the curve connecting these two points. According to the definition of an arc, <note>arc MO</note> is a segment of the curve between <note>two points M and O</note> on the circle. There are two points <note>P and N</note> on the circle, and <note>arc PN</note> is a segment of the curve connecting these two points. According to the definition of an arc, <note>arc PN</note> is a segment of the curve between <note>two points P and N</note> on the circle.

Inscribed Angle Theorem

<note>Angle MQO</note> is the angle formed by two chords <note>MN and PO</note> intersecting within the circle, so <note>Angle MQO</note> is an inscribed angle, i.e., <note>Angle MQO = 1/2 (arc MO + arc PN)</note>.

Definition of Linear Pair of Angles

<note>Angle MQO and angle NQO have a common side QO, and their other sides MQ and NQ are extensions in opposite directions</note>, so angle MQO and angle NQO are adjacent supplementary angles, meaning the sum of the two adjacent angles is 180 degrees.


Question:
Find the measure of minor arc $\overset{\frown}{WY}$ .

Write your answer as a whole number or a decimal.

m $\overset{\frown}{WY}$ = $\Box$ °

Answer: m \$\overset{\frown}{WY}\$ =13°

1. Given ∠VXZ = 50°, and from the diagram, we see that m ∠VXZ is formed by the secants XV and XZ intersecting at point X outside the circle. The measure of arc VZ in the diagram is 113°.

2. According to the external angle theorem of the circle, the measure of an external angle is equal to half the difference of the measures of the two intercepted arcs: m ∠VXZ = 1/2(m arc VZ - m arc WY).

3. Substituting the given conditions, we get the equation: m ∠VXZ = 1/2(m arc VZ - m arc WY), that is, 50° = 1/2(113° - m arc WY).

4. Solving the equation through simple algebra: multiplying both sides of the equation by 2, we get 100° = 113° - m arc WY.

5. Solving the equation: moving 113° to the other side of the equation, we get 100° - 113° = -m arc WY, simplifying to -13° = -m arc WY.

6. Eliminating the negative sign, we get m arc WY = 13°.

7. Through the above reasoning, the final answer is 13°.

External Angle Theorem of a Circle

In <note>circle O</note>, <note>angle VXZ is the external angle of the circle</note>, arc VZ and arc WY are <note>the two arcs subtended by angle VXZ</note>. According to the External Angle Theorem of a Circle, <note>angle VXZ is equal to half the difference of the degrees of arc VZ and arc WY</note>, that is, <note>angle VXZ = (degrees of arc VZ - degrees of arc WY) / 2</note>.

Secant Line

A straight line that intersects a circle at two distinct points is called a secant line of the circle.

In the figure of this problem, <note>line XV intersects the circle at two points, V and W</note>; <note>line XZ intersects the circle at two points, Z and Y</note>. According to the definition of a secant line, <note>line XV intersects the circle at two distinct points V and W, so line XV is a secant line of the circle</note>; similarly, <note>line XZ intersects the circle at two distinct points Z and Y, so line XZ is a secant line of the circle</note>.


Question:
What is m $\angle $ Q?

m $\angle $ Q= $\Box$ °

Answer: m \$\angle \$ Q=95°

1. Given that Q, R, S, T are on the circle, it is known that quadrilateral QRST is a cyclic quadrilateral.

2. According to the 'Opposite Angles of Cyclic Quadrilateral Theorem' (Corollary 3 of the Inscribed Angle Theorem), we can deduce that ∠S and ∠Q are supplementary, i.e., ∠S + ∠Q = 180°.

3. It is given in the problem that ∠S = 85°.

4. Therefore, we can set up the equation: ∠S + ∠Q = 180°, substituting the given condition into the equation, we get 85° + ∠Q = 180°.

5. Solving this equation, we find ∠Q = 180° - 85°.

6. Calculating 180° - 85°, we get ∠Q = 95°.

7. Through the above reasoning, the final answer is ∠Q = 95°.

Cyclic Quadrilateral

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

<note>The four vertices Q, R, S, and T of quadrilateral QRST are all on the same circle</note>. This circle is called the circumcircle of quadrilateral QRST. Therefore, <note>quadrilateral QRST is a cyclic quadrilateral</note>. According to <note>the properties of cyclic quadrilaterals</note>, it can be concluded that <note>the sum of opposite angles is equal to 180 degrees</note>, that is, <note>angle QRS + angle QTS = 180 degrees, angle RQT + angle RST = 180 degrees</note>.

In the diagram of this problem, <note>the vertices Q, R, S, T of the cyclic quadrilateral QRST are all on the circle</note>. According to <note>the Diagonal Supplementary Theorem for Cyclic Quadrilateral</note>, <note>the sum of each pair of opposite angles of the quadrilateral QRST is equal to 180°</note>. Specifically, <note>∠QRS + ∠QTS = 180°</note>; <note>∠RQT + ∠RST = 180°</note>.


Question:
Find the measure of  $\angle $ JNM.

Write your answer as a whole number or a decimal.

m $\angle $ JNM= $\Box$ °

Answer: m \$\angle \$ JNM=117.5°

1. Given chords \\overline{JK} and \\overline{LM} intersect at point N, find the degree measure of \\angle JNM. \\n\\n2. According to the theorem of angles inside a circle, for intersecting chords within a circle, the angle at their intersection point (such as \\angle MNK) is equal to half the sum of the measures of the arcs subtended by the angle (\\overset{\\frown}{LJ} and \\overset{\\frown}{KM}): \\angle MNK = \\frac{1}{2} (m \\overset{\\frown}{LJ} + m \\overset{\\frown}{KM}). \\n\\n3. Applying the above condition to the given arc measures, we get: m \\angle MNK = \\frac{1}{2} (52° + 73°) = 62.5°. \\n\\n4. Recognize that \\angle JNM and \\angle MNK form a straight angle. According to the definition of a straight angle, the total measure is 180°, so: m \\angle JNM = 180° - m \\angle MNK. \\n\\n5. Thus: m \\angle JNM = 180° - 62.5° = 117.5°. \\n\\n6. Through the above reasoning, the final answer is m \\angle JNM = 117.5°.

Definition of Chord

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

In the figure of this problem, in the circle, <note>points J and K are any two points on the circle</note>, <note>the line segment \overline{JK} connects these two points</note>, so <note>the line segment \overline{JK} is a chord of the circle</note>. Similarly, <note>points L and M are any two points on the circle</note>, <note>the line segment \overline{LM} connects these two points</note>, so <note>the line segment \overline{LM} is a chord of the circle</note>.

Inscribed Angle Theorem

In the figure of this problem, <note>angle MNK is the angle formed by the intersection of chords KJ and LM within the circle</note>, so <note>angle MNK is an inscribed angle</note>, that is, <note>angle MNK measure = 1/2 (arc KM measure + arc LJ measure)</note>.

Definition of Straight Angle

<note>Ray JN and ray NK are on the same straight line</note>, <note>angle JNK is a straight angle</note>, with a measure of 180 degrees.


Question:
△DEF and △JKL are shown below.

Which statement is true?

- △DEF is similar to △JKL.
- △DEF is not similar to △JKL.
- There is not enough information to determine whether the triangles are similar.

Answer: - △DEF is not similar to △JKL.

1. Given ∠E = 90°, ∠F = 32°, according to the triangle angle sum theorem, the sum of all interior angles is 180°.

2. According to the triangle angle sum theorem, let ∠D be x, then x + ∠E + ∠F = 180°.

3. Substitute the known values, we get x + 90° + 32° = 180°.

4. Solve the equation x + 122° = 180°, we get x = 58°, i.e., ∠D = 58°.

5. Therefore, the interior angles of △DEF are ∠D = 58°, ∠E = 90°, ∠F = 32°.

6. Observe △JKL, where ∠J = 52°, ∠K = 90° is known.

7. If △DEF and △JKL are similar, according to the similarity theorem (AA), the corresponding interior angles must be equal.

8. Check the corresponding interior angles: ∠J and any angle in △DEF are not 52°, so there cannot be two corresponding interior angles equal.

9. In summary, since there are no two corresponding interior angles equal, △DEF cannot be similar to △JKL.

10. Through the above reasoning, the final answer is △DEF is not similar to △JKL.

Definition of Triangle

In the diagram of this problem, there are two triangles, namely <note>△DEF</note> and <note>△JKL</note>. Each triangle is composed of three non-collinear points and their connecting line segments. <note>△DEF consists of vertices D, E, F and their connecting line segments DE, EF, FD</note>, corresponding to the interior angles <note>∠DEF</note>, <note>∠EFD</note>, <note>∠FDE</note>. <note>△JKL consists of vertices J, K, L and their connecting line segments JK, KL, LJ</note>, corresponding to the interior angles <note>∠JKL</note>, <note>∠KLJ</note>, <note>∠LJK</note>.

Similarity Theorem for Triangles (AA)

In the problem diagram, to determine whether △DEF and △JKL are similar, the AA condition must be met, which means two corresponding interior angles are equal. It is found that <note>∠E=90° and ∠K=90° are equal</note>, but the other angle does not meet the condition, for example, <note>∠F=32° and ∠L is not equal to 52°</note>, therefore these two triangles are not similar.

Triangle Angle Sum Theorem

<note>In triangle DEF, angle D, angle E, and angle F are the three interior angles of triangle DEF</note>, according to the Triangle Angle Sum Theorem, <note>angle D + angle E + angle F = 180°</note>. Similarly, <note>in triangle JKL, angle J, angle K, and angle L are the three interior angles of triangle JKL</note>, according to the Triangle Angle Sum Theorem, <note>angle J + angle K + angle L = 180°</note>.


Question:
UV is tangent to the circle at V. Find the measure of minor arc $\overset{\frown}{TV}$ .

Write your answer as a whole number or a decimal.

m $\overset{\frown}{TV}$ = $\Box$ °

Answer: m \$\overset{\frown}{TV}\$ =58°

1. Given that UV is a tangent line and it touches the circle at point V. From the figure, we get ∠TUV = 90°.

2. Given that US is a secant line, and US intersects the tangent line UV at point U, and arcs TV and SV form the corresponding angle ∠TUV = 90° inside the circle.

3. According to the exterior angle theorem of a circle, the angle at the intersection of the tangent and secant lines (∠TUV) is equal to half the difference of the intercepted arcs, i.e., m∠TUV = 1/2(arc SV - arc TV).

4. It is given in the problem that arc SV = 238°. From previous information, we know ∠TUV = 90°.

5. Substitute the known data into the exterior angle theorem equation: 90° = 1/2(238° - arc TV).

6. Multiply both sides of the equation by 2 to get 180° = 238° - arc TV.

7. Rearrange the equation to get arc TV = 238° - 180°.

8. Calculate to get arc TV = 58°.

9. Through the above reasoning, the final answer is 58°.

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.

In the figure of this problem, <note>the circle and the line UV have only one common point V</note>, <note>this common point is called the point of tangency</note>. Therefore, <note>the line UV is the tangent to the circle</note>.

Definition of Arc

<note>There are three points T, S, and V on the circle</note>, <note>Arc TV is a segment of the curve connecting point T and point V</note>, <note>Arc SV is a segment of the curve connecting point S and point V</note>. According to the definition of arc, <note>Arc TV is a segment of the curve between two points T and V on the circle</note>, <note>Arc SV is a segment of the curve between two points S and V on the circle</note>.

External Angle Theorem of a Circle

<note>Angle TUV is the external angle of the circle</note>, <note>arc TV and arc SV are the two arcs that angle TUV subtends</note>. According to the External Angle Theorem of a Circle, <note>angle TUV is equal to half the difference between the degree measures of arc TV and arc SV</note>, i.e., angle TUV = (degree measure of arc SV - degree measure of arc TV) / 2.


Question:
What is m $\angle $ U?

m $\angle $ U= $\Box$ °

Answer: m \$\angle \$ U=72°

1. It is known that quadrilateral TUVW is a cyclic quadrilateral, and the problem provides the angle ∠TWV = 108°.

2. According to the properties of cyclic quadrilaterals, the opposite angles of the quadrilateral are supplementary, i.e., ∠TWV + ∠TUV = 180°.

3. Since ∠TWV = 108°, substituting into the supplementary condition gives ∠TUV + 108° = 180°.

4. Solving the equation: ∠TUV + 108° = 180°, we get ∠TUV = 180° - 108°.

5. Calculating, we get ∠TUV = 72°.

6. Through the above reasoning, the final answer is 72°.

Cyclic Quadrilateral

In the diagram of this problem, <note>the four vertices T, U, V, and W of quadrilateral TUVW are all on the same circle</note>. This <note>circle is called the circumcircle of quadrilateral TUVW</note>. Therefore, quadrilateral TUVW is a cyclic quadrilateral. According to the properties of cyclic quadrilaterals, it can be concluded that <note>the sum of opposite angles is equal to 180 degrees</note>, that is, <note>angle ∠TWV + angle ∠TUV = 180 degrees</note>, <note>angle ∠UVW + angle ∠UTW = 180 degrees</note>.

<note>Quadrilateral TUVW is a cyclic quadrilateral</note>, according to the Diagonal Supplementary Theorem for Cyclic Quadrilateral, the sum of each pair of opposite angles of quadrilateral TUVW is equal to 180°. Specifically, <note>∠TUV + ∠TWV = 180°</note>; <note>∠UTW + ∠UVW = 180°</note>.


Question:
What is m $\angle $ I?

m $\angle $ I= $\Box$ °

Answer: m \$\angle \$ I=100°

1. According to the inscribed quadrilateral, quadrilateral GHIJ is an inscribed quadrilateral.

2. According to (corollary 3 of the inscribed angle theorem) the supplementary angles theorem of inscribed quadrilateral, when a quadrilateral is inscribed in a circle, its opposite angles are supplementary, that is, their sum is 180°.

3. Let the angles of opposite vertices G and I be ∠G and ∠I respectively. According to the above property, we have: ∠G + ∠I = 180°.

4. Given ∠G = 80°, substitute it into the supplementary angle relationship to get: 80° + ∠I = 180°.

5. Solve the equation 80° + ∠I = 180°, subtract 80° from both sides to get: ∠I = 180° - 80°.

6. Calculate to get: ∠I = 100°.

7. Therefore, the measure of ∠I is 100°.

8. Through the above reasoning, the final answer is 100°.

<note>The four vertices G, H, I, and J of quadrilateral GHIJ are on the same circle</note>. <note>This circle is called the circumcircle of quadrilateral GHIJ</note>. Therefore, quadrilateral GHIJ is a cyclic quadrilateral. According to the properties of cyclic quadrilaterals, it can be concluded that <note>the sum of opposite angles is equal to 180 degrees</note>, that is, <note>angle ∠G + angle ∠I = 180 degrees</note>, <note>angle ∠H + angle ∠J = 180 degrees</note>.

<note>In the cyclic quadrilateral GHIJ, the vertices G, H, I, and J 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 GHIJ is equal to 180°</note>. Specifically, <note>∠GHI + ∠GJI = 180°</note>; <note>∠HGJ + ∠HIJ = 180°</note>.


Question:
What is m $\angle $ S?

m $\angle $ S= $\Box$ °

Answer: m \$\angle \$ S=87°

1. Given that quadrilateral PQRS is a cyclic quadrilateral.

2. According to (corollary 3 of the inscribed angle theorem) the opposite angles of a cyclic quadrilateral are supplementary: the opposite angles of the quadrilateral are supplementary, i.e., ∠Q and ∠S are supplementary angles, which means ∠Q + ∠S = 180°.

3. Given that the measure of ∠Q is 93°, substituting the given condition, we can obtain the equation: ∠Q + ∠S = 180°, i.e., 93° + ∠S = 180°.

4. By calculation, ∠S = 180° - 93°.

5. The calculation yields ∠S = 87°.

6. Therefore, through the above reasoning, the final answer is 87°.

<note>The four vertices P, Q, R, and S of quadrilateral PQRS lie on the same circle</note>. This circle is called <note>the circumcircle of quadrilateral PQRS</note>. Therefore, <note>quadrilateral PQRS is a cyclic quadrilateral</note>. According to the properties of cyclic quadrilaterals, the sum of opposite angles is equal to 180 degrees, i.e., <note>angle PQR + angle PSR = 180 degrees</note>, <note>angle QPS + angle QRS = 180 degrees</note>.

In the figure of this problem, in the cyclic quadrilateral <note>PQRS</note>, the vertices <note>P, Q, R, S are all on the circle</note>. According to the <note>Diagonal Supplementary Theorem for Cyclic Quadrilateral</note>, the sum of each pair of opposite angles of quadrilateral PQRS is equal to 180°. Specifically, <note>∠PQR + ∠PSR = 180°</note>; <note>∠QPS + ∠QRS = 180°</note>.


Question:
What is m $\angle $ H?

m $\angle $ H= $\Box$ °

Answer: m \$\angle \$ H=104°

1. According to the inscribed quadrilateral, quadrilateral HIJK is an inscribed quadrilateral. According to the (corollary 3 of the inscribed angle theorem) supplementary angles theorem of inscribed quadrilateral, if a quadrilateral is an inscribed quadrilateral, then each pair of its opposite angles are supplementary angles.

2. According to the above theorem, in the inscribed quadrilateral HIJK, angle H and angle J are supplementary angles, so ∠H + ∠J = 180°.

3. Given ∠J = 76°, substitute it into the supplementary angle equation, we get ∠H + 76° = 180°.

4. Solve this equation, subtract 76°, we get ∠H = 180° - 76° = 104°.

5. Through the above reasoning, the final answer is 104°.

In the diagram of this problem, <note>the four vertices H, I, J, and K of quadrilateral HIJK are all on the same circle</note>. This circle is called <note>the circumcircle of quadrilateral HIJK</note>. Therefore, <note>quadrilateral HIJK is a cyclic quadrilateral</note>. According to the properties of cyclic quadrilaterals, it can be concluded that <note>the sum of the opposite angles is 180 degrees, i.e., angle H + angle J = 180 degrees, angle I + angle K = 180 degrees</note>.

In the <note>cyclic quadrilateral HIJK</note>, <note>the vertices H, I, J, K of the quadrilateral are all on the circle</note>. According to the <note>Diagonal Supplementary Theorem for Cyclic Quadrilateral</note>, <note>the sum of each pair of opposite angles of quadrilateral HIJK is equal to 180°</note>. Specifically, <note>∠H + ∠J = 180°</note>; <note>∠I + ∠K = 180°</note>.


Question:
MN is tangent to the circle at N. Find the measure of  $\angle $ LMN.

Write your answer as a whole number or a decimal.

m $\angle $ LMN= $\Box$ °

Answer: m \$\angle \$ LMN=39°

1. Let circle O be given, and it is known that MN is the tangent to the circle at point N. MK is the secant line intersecting MN at M, and K and L are two points on the circle.

2. According to the exterior angle theorem of the circle, the angle ∠LMN satisfies the formula: ∠LMN = 1/2 (degree of arc KN - degree of arc LN).

3. Since the degree of arc KN = 152° and the degree of arc LN = 74°, according to the above formula, ∠LMN = 1/2 (152° - 74°).

4. Calculate the expression from the previous step: 1/2 (152° - 74°) = 1/2 × 78° = 39°.

5. Through the above reasoning, the final answer is 39°.

Definition of Tangent to a Circle

In the figure of this problem, <note>the circle and the line MN have only one common point N</note>, this common point is called the point of tangency. Therefore, <note>the line MN is the tangent to the circle</note>.

Secant Line

Original text: <note>Line MK intersects the circle at two points, namely point K and point L</note>. According to the definition of a secant line, <note>line MK intersects the circle at two distinct points</note>, so <note>line MK is a secant line of the circle</note>.

Definition of Arc

<note>There are two points K and N on the circle, and arc KN is a segment of the curve connecting these two points</note>; <note>There are also two points L and N on the circle, and arc LN is a segment of the curve connecting these two points</note>. According to the definition of arc, <note>arc KN is a segment of the curve between two points K and N on the circle</note>, <note>arc LN is a segment of the curve between two points L and N on the circle</note>. <note>The degree of arc KN is 152°</note>, <note>The degree of arc LN is 74°</note>.

External Angle Theorem of a Circle

<note>In circle O, ∠LMN is the external angle of the circle</note>, <note>arc KN and arc LN are the two arcs subtended by ∠LMN</note>. According to the External Angle Theorem of a Circle, <note>∠LMN is equal to half the difference between the degree measures of arc KN and arc LN</note>, that is, <note>∠LMN = (degree measure of arc KN - degree measure of arc LN) / 2</note>.


Question:
What is m $\angle $ K?

m $\angle $ K= $\Box$ °

Answer: m \$\angle \$ K=91°

1. According to the inscribed quadrilateral, it is concluded that quadrilateral HIJK is an inscribed quadrilateral, and the vertices H, I, J, K are all on the circle.

2. According to the (corollary 3 of the inscribed angle theorem) opposite angles of an inscribed quadrilateral are supplementary, the opposite angles of an inscribed quadrilateral are supplementary, that is, the sum of the opposite angles is equal to 180°.

3. In this problem, ∠I and ∠K are opposite angles, so ∠I + ∠K = 180°.

4. Given ∠I = 89°, substituting it in gives: 89° + ∠K = 180°.

5. From the equation 89° + ∠K = 180°, subtract 89° from both sides to get ∠K = 180° - 89°.

6. Calculate the right-hand side to get ∠K = 91°.

7. Through the above reasoning, the final answer is ∠K = 91°.

Cyclic Quadrilateral

In the figure of this problem, the four vertices H, I, J, and K of quadrilateral HIJK are all on the same circle. This circle is called the circumcircle of quadrilateral HIJK. Therefore, quadrilateral HIJK is a cyclic quadrilateral. According to the properties of cyclic quadrilaterals, it can be concluded that the sum of the opposite angles is equal to 180 degrees, that is, angle I + angle K = 180 degrees, angle H + angle J = 180 degrees.

In the diagram of this problem, <note>cyclic quadrilateral HIJK</note>, <note>the vertices H, I, J, and K of the quadrilateral are on the circle</note>. According to <note>Diagonal Supplementary Theorem for Cyclic Quadrilateral</note>, the sum of each pair of opposite angles of quadrilateral HIJK is equal to 180°. Specifically, <note>angle I + angle K = 180 degrees, angle H + angle J = 180 degrees</note>.


Question:
What is m $\angle $ S?

m $\angle $ S= $\Box$ °

Answer: m \$\angle \$ S=63°

1. Observing the figure, we can see that quadrilateral PQRS is a cyclic quadrilateral.

2. According to the (Inscribed Angle Theorem Corollary 3) Cyclic Quadrilateral Opposite Angles Theorem, this theorem states that if a quadrilateral is cyclic, then the sum of its opposite angles is 180 degrees.

3. Applying this theorem to quadrilateral PQRS, i.e., ∠PQR and ∠PSR are opposite angles, according to the theorem we know ∠PQR + ∠PSR = 180°.

4. Given ∠PQR = 117°, substitute into the equation from the previous step: 117° + ∠PSR = 180°.

5. Solve the equation to get: ∠PSR = 180° - 117° = 63°.

6. Through the above reasoning, the final answer is 63°.

<note>The four vertices P, Q, R, and S of quadrilateral PQRS all lie on the same circle</note>. This circle is called the <note>circumcircle of quadrilateral PQRS</note>. Therefore, <note>quadrilateral PQRS is a cyclic quadrilateral</note>. According to the properties of a cyclic quadrilateral, it can be concluded that <note>the sum of the opposite angles is equal to 180 degrees</note>, i.e., <note>angle PQR + angle PSR = 180 degrees</note>.

In the cyclic quadrilateral PQRS, the vertices P, Q, R, and S are all on the circle. According to the Diagonal Supplementary Theorem for Cyclic Quadrilateral, the sum of each pair of opposite angles in quadrilateral PQRS equals 180°. Specifically, ∠PQR + ∠PSR = 180°; given ∠PQR = 117°, the theorem can be used to find ∠PSR.


Question:
△CDE and △FGH are shown below.

Which statement is true?

- △CDE is similar to △FGH.
- △CDE is not similar to △FGH.
- There is not enough information to determine whether the triangles are similar.

Answer: - △CDE is not similar to △FGH.

1. In △CDE, it is known that ∠C = 48° and ∠E = 80°.

2. According to the triangle angle sum theorem, the sum of the three angles in a triangle is 180°, that is, ∠C + ∠D + ∠E = 180°.

3. Substituting the known values, we get 48° + ∠D + 80° = 180°.

4. Simplifying the equation, we get ∠D + 128° = 180°.

5. Solving the equation, we get ∠D = 52°, so the three angles of △CDE are 48°, 52°, and 80°.

6. In △FGH, it is known that ∠F = 48° and ∠G = 62°.

7. Similarly, according to the triangle angle sum theorem, we get ∠F + ∠G + ∠H = 180°.

8. Substituting the known values, we get 48° + 62° + ∠H = 180°.

9. Simplifying the equation, we get ∠H + 110° = 180°.

10. Solving the equation, we get ∠H = 70°, so the three angles of △FGH are 48°, 62°, and 70°.

11. Comparing the interior angles of △CDE and △FGH, we find that ∠D and ∠G are not equal (52° ≠ 62°), so the corresponding interior angles of the two triangles are not all equal.

12. According to the similarity theorem (AA), if two triangles are similar, their corresponding angles are equal.

13. Since the corresponding interior angles of △CDE and △FGH are not completely equal, △CDE is not similar to △FGH.

14. Through the above reasoning, the final conclusion is that △CDE is not similar to △FGH.

Triangle Angle Sum Theorem

In the figure of this problem, <note>in triangle CDE, angle C, angle D, and angle E are the three interior angles of triangle CDE</note>, according to the Triangle Angle Sum Theorem, <note>angle C + angle D + angle E = 180°</note>. Similarly, in <note>triangle FGH, angle F, angle G, and angle H are the three interior angles of triangle FGH</note>, according to the Triangle Angle Sum Theorem, <note>angle F + angle G + angle H = 180°</note>.

Similarity Theorem for Triangles (AA)

<note>In the figure of this problem, in triangles CDE and FGH, if angle C is equal to angle F, and angle E is equal to angle H</note>, then triangle CDE is not similar to triangle FGH. <note>According to the Similarity Theorem for Triangles (AA), the corresponding angles of the two triangles are not completely equal</note>, therefore △CDE and △FGH are not similar.


Question:
△DEF and △JKL are shown below.

Which statement is true?

- △DEF is similar to △JKL.
- △DEF is not similar to △JKL.
- There is not enough information to determine whether the triangles are similar.

Answer: - △DEF is not similar to △JKL.

1. Given in △DEF, ∠E is 90°, ∠F is 32°. We need to solve for the angle ∠D.

2. According to the triangle angle sum theorem, the sum of the angles in a triangle is 180°, we get ∠D + ∠E + ∠F = 180°.

3. Substitute the known values: ∠D + 90° + 32° = 180°.

4. Solve this equation to get ∠D = 180° - 122° = 58°.

5. Therefore, the three interior angles of △DEF are ∠D = 58°, ∠E = 90°, ∠F = 32°.

6. For △JKL, it is known that ∠K = 90° and ∠J = 52°.

7. If △DEF is similar to △JKL, according to the similarity theorem (AA), we need to check if the corresponding angles are equal.

8. Check the angles of △JKL corresponding to the angles of △DEF: For ∠E = 90°, corresponding to ∠K = 90°, they correspond; but ∠F = 32° does not correspond to ∠J = 52°.

9. Therefore, none of the angles of △DEF correspond to another angle of △JKL, so △DEF is not similar to △JKL.

10. After the above reasoning, the final answer is △DEF is not similar to △JKL.

Definition of Triangle

<note>Triangle DEF is a geometric figure consisting of three non-collinear points D, E, F and their connecting line segments DE, EF, FD</note>.<note>Points D, E, F are the three vertices of the triangle</note>,<note>Line segments DE, EF, FD are the three sides of the triangle</note>.<note>Triangle JKL is a geometric figure consisting of three non-collinear points J, K, L and their connecting line segments JK, KL, LJ</note>.<note>Points J, K, L are the three vertices of the triangle</note>,<note>Line segments JK, KL, LJ are the three sides of the triangle</note>.

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>∠DEF is a geometric figure composed of two rays DE and EF</note>, which have a <note>common endpoint E</note>. This <note>common endpoint E is called the vertex of ∠DEF</note>, and the rays DE and EF are called <note>the sides of ∠DEF</note>. Similarly, <note>∠JKL is a geometric figure composed of two rays JK and KL</note>, which have a <note>common endpoint K</note>. This <note>common endpoint K is called the vertex of ∠JKL</note>, and the rays JK and KL are called <note>the sides of ∠JKL</note>. <note>∠DFE is a geometric figure composed of two rays DF and EF</note>, which have a <note>common endpoint F</note>. This <note>common endpoint F is called the vertex of ∠DFE</note>, and the rays DF and EF are called <note>the sides of ∠DFE</note>. <note>∠EDF is a geometric figure composed of two rays DE and DF</note>, which have a common endpoint.

Triangle Angle Sum Theorem

In this problem diagram, the three interior angles of <note>triangle DEF</note>, <note>angle D</note>, <note>angle E</note>, and <note>angle F</note>, sum up to 180° according to the Triangle Angle Sum Theorem, <note>angle D + angle E + angle F = 180°</note>. Similarly, in <note>triangle JKL</note>, the three interior angles of <note>angle J</note>, <note>angle K</note>, and <note>angle L</note>, sum up to 180° according to the Triangle Angle Sum Theorem, <note>angle J + angle K + angle L = 180°</note>.

Similarity Theorem for Triangles (AA)

<note>△DEF and △JKL need to be similar</note>, then it requires <note>∠D=∠J</note>, <note>∠E=∠K</note>, <note>∠F=∠L</note>. Upon inspection, although <note>∠E and ∠K are both 90 degrees</note>, <note>∠F and ∠J do not correspond</note>, that is <note>32 degrees is not equal to 52 degrees</note>, therefore <note>△DEF and △JKL are not similar</note>.


Question:
What is m $\angle $ I?

m $\angle $ I= $\Box$ °

Answer: m \$\angle \$ I=33°

1. According to the problem statement, segment IJ is the diameter of the circle. Based on (corollary 2 of the inscribed angle theorem), the inscribed angle subtended by the diameter is a right angle. Therefore, ∠IHJ is a right angle.

2. Since ∆HIJ is a right triangle, based on the property of complementary acute angles in a right triangle, the sum of the two remaining angles (i.e., the acute angles) is 90°. Therefore, ∠HIJ + ∠IJH = 90°.

3. Given ∠IJH = 57°, substituting into the above equation gives: 57° + ∠HIJ = 90°.

4. Solving the equation 57° + ∠HIJ = 90°, we get ∠HIJ = 90° - 57° = 33°.

5. Through the above reasoning, the final answer is 33°.

Definition of Right Triangle

In the diagram of this problem, in triangle HIJ, <note>angle IHJ is a right angle (90 degrees)</note>, therefore triangle HIJ is a right triangle. <note>Side HI and side HJ are the legs</note>, <note>side IJ is the hypotenuse</note>.

Complementary Acute Angles in a Right Triangle

In a right triangle, the sum of the two non-right angles is 90°.

<note>In right triangle HIJ</note>, <note>angle IHJ is a right angle (90 degrees)</note>, angle HIJ and angle HJI are the two acute angles other than the right angle. According to the property of complementary acute angles in a right triangle, <note>the sum of angle HIJ and angle HJI is 90 degrees</note>, i.e., <note>angle HIJ + angle HJI = 90°</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.

In the diagram of this problem, in the circle, the <note>diameter IJ</note> subtends a <note>right angle (90 degrees) at the circumference IHJ</note>.


Question:
What is m $\angle $ G?

m $\angle $ G= $\Box$ °

Answer: m \$\angle \$ G=39°

1. Given that \( \overline{GH} \) is the diameter of the circle, according to (Corollary 2 of the Inscribed Angle Theorem) the inscribed angle subtended by the diameter is a right angle, the inscribed angle \( \angle GFH \) is a right angle, i.e., \( 90^\circ \).

2. From the conclusion of the first step, \( \triangle FGH \) is a right triangle with the right angle being \( \angle GFH \).

3. According to the Triangle Sum Theorem, the sum of the interior angles of any triangle is \( 180^\circ \), therefore in \( \triangle FGH \), we have \( \angle FGH + \angle GHF + \angle GFH = 180^\circ \).

4. Substitute the known values, \( \angle GFH = 90^\circ \) and \( \angle FHG = 51^\circ \), then: \( 51^\circ + \angle G + 90^\circ = 180^\circ \).

5. Solve this equation, we get \( \angle G = 180^\circ - 90^\circ - 51^\circ = 39^\circ \).

6. Through the above reasoning, the final answer is \( \angle G = 39^\circ \).

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 the circle, <note>the vertex F of angle \( \angle GFH \) is on the circumference</note>, and the two sides of angle \( \angle GFH \) <note>intersect the circle at points G and H</note>. Therefore, <note>angle \( \angle GFH \) is an inscribed angle</note>.

Definition of Right Triangle

In <note>\( \triangle FGH \)</note>, <note>\( \angle GFH \) is a right angle (90 degrees)</note>, so <note>\( \triangle FGH \)</note> is a right triangle. Side <note>\( FG \)</note> and side <note>\( FH \)</note> are the legs, and side <note>\( GH \)</note> is the hypotenuse.

<note>The inscribed angle GFH subtended by the diameter HG is a right angle (90 degrees)</note>. (Or <note>The inscribed angle GFH is 90 degrees, so the chord HG it subtends is a diameter</note>.)

Triangle Angle Sum Theorem

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

<note>\( \triangle FGH \)</note> is a triangle, therefore it satisfies the Triangle Angle Sum Theorem, that is <note>\( \angle FGH + \angle HGF + \angle GFH = 180^\circ \)</note>. Substituting the known values <note>\( \angle GFH = 90^\circ \)</note> and <note>\( \angle FGH = 51^\circ \)</note>, we solve for <note>\( \angle HGF = 39^\circ \)</note>.


Question:
What is m $\angle $ R?

m $\angle $ R= $\Box$ °

Answer: m \$\angle \$ R=53°

1. Given that segment PR is the diameter of the circle, according to the Inscribed Angle Theorem, the inscribed angle subtended by the diameter is a right angle. Therefore, ∠PQR is a right angle, i.e., m∠PQR = 90°.

2. Triangle PQR is a right triangle, so according to the Triangle Angle Sum Theorem, the sum of the three interior angles of the triangle is 180°, hence m∠PQR + m∠QRP + m∠RPQ = 180°.

3. From the first two steps, we know that m∠PQR = 90°, and it is given that m∠RPQ = 37°, therefore we can find m∠QRP.

4. Based on the angle sum equation from step 2 and the known angles, we get 90° + 37° + m∠QRP = 180°.

5. Solving the above equation: m∠QRP = 180° - 90° - 37° = 53°.

6. Through the above reasoning, the final answer is 53°.

Definition of Right Triangle

<note>Angle PQR is a right angle (90 degrees)</note>, therefore triangle PQR is a right triangle. Side PR and side QR are the legs, and side PQ is the hypotenuse.

Inscribed Angle Theorem

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

<note>Points P, Q, R are on the circle</note>, the central angle corresponding to arc PR and arc PQ is <note>∠POR</note>, and the inscribed angle is <note>∠PQR</note>. According to the Inscribed Angle Theorem, <note>∠PQR</note> is equal to half of the central angle <note>∠POR</note> corresponding to the arc <note>PR</note>, that is, <note>∠PQR = 1/2 ∠POR</note>.

Triangle Angle Sum Theorem

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


Question:
What is m $\angle $ J?

m $\angle $ J= $\Box$ °

Answer: m \$\angle \$ J=57°

1. According to the definition of diameter, line JK is the diameter of the circle; according to the definition of the inscribed angle, ∠JIK is an inscribed angle; according to (corollary 2 of the inscribed angle theorem) the inscribed angle subtended by the diameter is a right angle, it is concluded that ∠JIK is a right angle, i.e., ∠JIK = 90.

2. According to the definition of a right triangle, △IJK is a right triangle, and ∠JIK = 90. According to the triangle sum theorem, ∠JIK + ∠IKJ + ∠IJK = 180°, i.e., ∠IJK + ∠IKJ = 180° - 90° = 90°.

3. Given ∠IKJ = 33°, then ∠IJK + 33° = 90.

4. Solving yields ∠IJK = 90° - 33° = 57°.

5. Through the above reasoning, the final answer is ∠J = 57°.

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>JK is the diameter</note>, connecting the <note>center of the circle</note> and the points <note>J and K on the circumference</note>, with a length of <note>2 times the radius</note>, i.e., <note>JK = 2r</note>.

Definition of Inscribed Angle

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

Definition of Right Triangle

In the figure of this problem, in triangle JIK, <note>angle JIK is a right angle (90 degrees)</note>, therefore <note>triangle IJK is a right triangle</note>. <note>Sides IJ and IK are the legs</note>, <note>side JK is the hypotenuse</note>.

<note>Diameter JK</note> subtends a <note>right angle (90 degrees)</note> at the <note>circumference angle JIK</note>.

Triangle Angle Sum Theorem

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


Question:
What is m $\angle $ H?

m $\angle $ H= $\Box$ °

Answer: m \$\angle \$ H=52°

1. Given that line segment FH is the diameter of the circle, according to (Corollary 2 of the Inscribed Angle Theorem), the inscribed angle subtended by the diameter is a right angle, thus angle FGH is a right angle, i.e., ∠FGH = 90°.

2. Since angle FGH is a right angle, according to the definition of a right triangle, triangle FGH is a right triangle. According to the property of complementary acute angles in a right triangle, and ∠F and ∠H are complementary angles, the sum of the two angles is 90°.

3. Given ∠F = 38°, according to the property of complementary acute angles in a right triangle, ∠F + ∠H = 90°.

4. Substituting the given condition, 38° + ∠H = 90°.

5. Solving the equation, we get ∠H = 90° - 38° = 52°.

6. Through the above reasoning, the final answer is ∠H = 52°.

Definition of Right Triangle

<note>In triangle FGH, angle FGH is a right angle (90 degrees)</note>, therefore triangle FGH is a right triangle. <note>Side FG and side GH are the legs, side FH is the hypotenuse</note>.

Definition of Inscribed Angle

In the figure of this problem, in the circle, <note>the vertex of angle FGH (point G) is on the circumference</note>, <note>the two sides of angle FGH intersect the circle at points H and F respectively</note>. Therefore, <note>angle FGH is an inscribed angle</note>.

In the right triangle FGH, angle FGH is a right angle (90 degrees), angles F and H are the two acute angles other than the right angle. According to the property of complementary acute angles in a right triangle, the sum of angles F and H is 90 degrees, that is, angle F + angle H = 90°.

<note>The angle subtended by the diameter FH at the circumference, angle FGH, is a right angle (90 degrees)</note>.

Definition of Diameter

<note>In the figure of this problem, in the circle, segment FH is the diameter of the circle</note>. Segment FH passes through the center of the circle, and both endpoints F and H are on the circle. According to the definition of diameter, <note>segment FH is the longest chord of the circle, with a length equal to 2 times the radius, i.e., FH = 2 * radius</note>.


Question:
What is m $\angle $ I?

m $\angle $ I= $\Box$ °

Answer: m \$\angle \$ I=52°

1. Given that line segment \overline{IK} is the diameter of the circle, points I, J, K are on the circle. Therefore, according to (Corollary 2 of the Inscribed Angle Theorem), the inscribed angle subtended by the diameter is a right angle, thus \angle IJK is a right angle, i.e., \angle IJK = 90°.

 2. In the right triangle \triangle IJK, according to the Triangle Sum Theorem, the sum of the three interior angles is 180°, thus \angle I + \angle IKJ + \angle IJK = 180°.

 3. Based on \angle IJK = 90° known from step 1, substituting gives \angle I + \angle IKJ + 90° = 180°.

 4. Based on the known condition \angle IKJ = 38°, substituting gives \angle I + 38° + 90° = 180°.

 5. Simplifying the equation gives \angle I + 128° = 180°.

 6. Subtracting 128° from both sides gives \angle I = 52°.

 7. Through the above reasoning, the final answer is \angle I = 52°.

Triangle Angle Sum Theorem

In the figure of this problem, the angles <note>\angle I</note>, <note>\angle IKJ</note>, and <note>\angle IJK</note> are the three interior angles of triangle <note>\triangle IJK</note>. According to the Triangle Angle Sum Theorem, <note>\angle I</note> + <note>\angle IKJ</note> + <note>\angle IJK</note> = 180°.

The angle subtended by the diameter <note>IK</note> at the circumference <note>IJK</note> is a <note>right angle (90 degrees)</note>.


Question:
The radius of a circle is 9 inches. What is the length of a 135° arc?

Give the exact answer in simplest form.
 $\Box$ 𝜋/ inches

Answer: 27/4𝜋 inches

1. Given the radius of the circle is 9 inches, according to the formula for the circumference of a circle C=2πr, we get the circumference C=2×π×9=18π inches.

2. The ratio of the length of an arc in a circle to the circumference is equal to the ratio of the central angle corresponding to the arc to the entire circle's 360°, the formula is: l = C * (m/360), where l is the arc length, C is the circumference, and m is the central angle.

3. The central angle given in the problem is 135°, substituting C=18π inches and m=135 into the arc length formula, we get l = 18π * (135/360).

4. Simplify the fraction inside the parentheses, 135/360 equals 3/8.

5. Substituting the simplified fraction into the arc length formula, we get l = 18π * (3/8) = (18 * 3/8)π.

6. Calculate the multiplication inside the parentheses, 18 * 3 = 54, then 54 divided by 8, we get the simplified result l = 27π/4 inches.

7. Through the above reasoning, the final answer is 27π/4 inches.

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 diagram of this problem, <note>in the circle, the point at the center of the diagram is the center of the circle</note>, <note>the radius is 9 inches</note>. All points in the diagram <note>that are 9 inches away from the center of the circle are on the circle</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>In the diagram of this problem, the radius of the circle r=9 inches</note>. According to the circumference formula of the circle, the circumference C is equal to 2π multiplied by the radius r, <note>i.e., C=2πr</note>. Therefore, <note>the circumference of the circle C=2π×9=18π inches</note>.

Arc Length Formula of a Circle

The arc length refers to the length of a segment of the circumference of a circle. It can be calculated using the formula 𝓁 = C × (m/360), where m is the measure of the central angle in degrees.

<note>In the figure of this problem</note>, <note>in the circle</note>, the degree of the central angle is <note>135 degrees</note>, and the circumference is <note>C</note>. According to the arc length formula of a circle, the arc length can be expressed as 𝓁 = C × (m/360). Therefore, the arc length is equal to <note>the circumference 18π</note> multiplied by <note>the degree of the central angle 135</note> divided by <note>360 degrees</note>.


Question:
What is m $\angle $ F?

m $\angle $ F= $\Box$ °

Answer: m \$\angle \$ F=37°

1. According to the given figure and the point G on the circumference and the diameter \overline{EF}, since \overline{EF} is the diameter, according to (Corollary 2 of the Inscribed Angle Theorem), the inscribed angle subtended by the diameter is a right angle: \angle EGF is a right angle. Therefore, △EFG is a right triangle.

2. In the right triangle △EFG, according to the property that the acute angles of a right triangle are complementary, \angle E and \angle F are complementary, that is, \angle E + \angle F = 90°.

3. Given \angle E = 53°, we can use the conclusion from the previous step: \angle E + \angle F = 90°, substituting the known condition to get 53° + \angle F = 90°.

4. Through the equation 53° + \angle F = 90°, we get \angle F = 90° - 53°.

5. Calculating, we get: \angle F = 37°.

6. Through the above reasoning, the final answer is that the measure of \angle F is 37°.

Definition of Right Triangle

In the figure of this problem, in triangle EFG, <note>angle EGF is a right angle (90 degrees)</note>, therefore triangle EFG is a right triangle. <note>Side EG and side GF are the legs</note>, <note>side EF is the hypotenuse</note>.

The angle subtended by <note>diameter EF</note> is a right angle (<note>90 degrees</note>). (Or in the figure of this problem, <note>the inscribed angle EGF</note> is <note>90 degrees</note>, so the chord it subtends, <note>EF</note>, is the diameter.)

In the figure of this problem, in the right triangle <note>EFG</note>, <note>angle EGF is a right angle (90 degrees)</note>, angle <note>E</note> and angle <note>F</note> are the two acute angles other than the right angle. According to the complementary acute angles property of a right triangle, <note>the sum of angle E and angle F is 90 degrees</note>, that is, <note>angle E + angle F = 90°</note>.


Question:
The radius of a circle is 1 mile. What is the length of a 45° arc?

Give the exact answer in simplest form.
 $\Box$ 𝜋/ miles

Answer: 1/4𝜋 miles

1. Given the radius of the circle r = 1 mile, we need to calculate the circumference of the circle. According to the formula for the circumference of a circle: C = 2πr, substituting the given r = 1 mile, we get the circumference of the circle C = 2π miles.

2. We need to calculate the arc length l, where the arc corresponds to a central angle of 45°. The formula for the arc length is: l = C * (θ/360°), where C is the circumference of the circle and θ is the central angle corresponding to the arc.

3. Substituting the given conditions and the result from the previous step into the arc length formula, we get l = 2π * (45°/360°).

4. Simplifying the above expression: l = 2π * (1/8) = π/4.

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

Definition of Radius

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

In the figure of this problem, <note>the radius of the circle r = 1 mile</note>, that is, <note>the length of the line segment from the center of the circle to any point on the circumference is 1 mile</note>. <note>The center of the circle is point O, and any point on the circumference is point A</note>, so <note>the line segment OA is the radius of the circle</note>, represented as <note>r = 1 mi</note>.

Definition of Arc

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

In the figure of this problem, there are two points <note>A and B</note> on the circle, and <note>arc AB</note> is a segment of the curve connecting these two points. According to the definition of an arc, <note>arc AB</note> is a segment of the curve between two points A and B on the circle, with <note>an angle of θ = 45°</note>.

Circumference Formula of Circle

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

Arc Length Formula of a Circle

In the problem diagram, <note>circumference C = 2π miles</note>, <note>central angle θ corresponding to the arc = 45°</note>, so the arc length l = 2π * (45° / 360°) = <note>π/4 miles</note>.