780 Chapter 12 Circles

An angle whose vertex is on the circle and whose sides are chords of the circle is an inscribed angle. An arc with endpoints on the sides of an inscribed angle, and its other points in the interior of the angle is an intercepted arc. In the diagram, inscribed ∠C intercepts AB¬.

Essential Understanding Angles formed by intersecting lines have a special relationship to the arcs the intersecting lines intercept. In this lesson, you will study arcs formed by inscribed angles. hsm11gmse_1203_t06873

AInterceptedarc

Inscribed angle

B C

Theorem 12-11 Inscribed Angle Theorem

The measure of an inscribed angle is half the measure of its intercepted arc.

m∠B = 12 mAC¬

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A

B

C

Objectives To find the measure of an inscribed angleTo find the measure of an angle formed by a tangent and a chord

Three high-school soccer players practice kicking goals from the points shown in the diagram. All three points are along an arc of a circle. Player A says she is in the best position because the angle of her kicks toward the goal is wider than the angle of the other players’ kicks. Do you agree? Explain.

Inscribed Angles12-3

Draw a large diagram and draw the angle each point makes with the goal posts.

Lesson Vocabulary

•inscribedangle•interceptedarc

LessonVocabulary

Player A

Player B

Player C

Player A

Player B

Player C

MATHEMATICAL PRACTICES

G-C.A.2 Identify and describe relationships among inscribed angles, radii, and chords. Also G-C.A.3, G-C.A.4

MP 1, MP 3, MP 4, MP 6

Common Core State Standards

Problem 1

Lesson 12-3 InscribedAngles 781

To prove Theorem 12-11, there are three cases to consider.

Below is a proof of Case I. You will prove Case II and Case III in Exercises 26 and 27.

Proof of Theorem 12-11, Case I

Given: }O with inscribed ∠B and diameter BC

Prove: m∠B = 12 m AC¬

Draw radius OA to form isosceles △AOB with OA = OB and, hence, m∠A = m∠B (Isosceles Triangle Theorem).

m∠AOC = m∠A + m∠B Triangle Exterior Angle Theorem

mAC¬ = m∠AOC Definition of measure of an arc

mAC¬ = m∠A + m∠B Substitute.

mAC¬ = 2m∠B Substitute and simplify.

12 mAC¬ = m∠B Divide each side by 2.

Using the Inscribed Angle Theorem

What are the values of a and b?

m∠PQT = 12 m PT¬ Inscribed Angle Theorem

60 = 12 a Substitute.

120 = a Multiply each side by 2.

m∠PRS = 12 m PS¬ Inscribed Angle Theorem

m∠PRS = 12 (m PT¬ + m TS¬ ) Arc Addition Postulate

b = 12 (120 + 30) Substitute.

b = 75 Simplify.

1. a. In }O, what is m∠A? b. What are m∠A, m∠B, m∠C, and m∠D?

c. What do you notice about the sums of the measures of the opposite angles in the quadrilateral in part (b)?

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OO

O

I: The center is on a side of the angle.

II: The center is inside the angle.

III: The center is outside the angle.

Proof

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A C

B

O

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R

S

TQ

P

30�60�

b�

a�

Got It?

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OA

Which variable should you solve for first?You know the inscribed angle that intercepts PT¬, which has the measure a. You need a to find b. So find a first.

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106� 100�

90�64�

A

B

C

D

Problem 2

782 Chapter 12 Circles

You will use three corollaries to the Inscribed Angle Theorem to find measures of angles in circles. The first corollary may confirm an observation you made in the Solve It.

Using Corollaries to Find Angle Measures

What is the measure of each numbered angle?

A B

∠1 is inscribed in a semicircle. ∠2 and the 38° angle intercept the By Corollary 2, ∠1 is a right angle, so same arc. By Corollary 1, the angles m∠1 = 90. are congruent, so m∠2 = 38.

2. In the diagram at the right, what is the measure of each numbered angle?

The following diagram shows point A moving along the circle until a tangent is formed.

From the Inscribed Angle Theorem, you know that in the first three diagrams m∠A is 12 mBC¬. As the last diagram suggests, this is also true when A and C coincide.

Corollaries to Theorem 12-11: The Inscribed Angle Theorem

Corollary 1Two inscribed angles that intercept the same arc are congruent.

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BA

Corollary 2An angle inscribed in a semicircle is a right angle.

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C

Corollary 3The opposite angles of a quadrilateral inscribed in a circle are supplementary.

You will prove these corollaries in Exercises 31–33.

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C

AB

D

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40�

1 70�

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70�

2

38�

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60�

2

1

3 80�

4

Got It?

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AB B

C C

A

AA

B B

C C

A

A

Is there too much information?Each diagram has more information than you need. Focus on what you need to find.

Problem 3

Lesson 12-3 InscribedAngles 783

Using Arc Measure

In the diagram, <SR

> is a tangent to the circle at Q. If mPMQ¬ = 212,

what is mjPQR?

12 mPMQ¬ = m∠PQS

12 (212) = m∠PQS Substitute.

106 = m∠PQS Simplify.

m∠PQS + m∠PQR = 180 Linear Pair Postulate

106 + m∠PQR = 180 Substitute.

m∠PQR = 74 Simplify.

3. a. In the diagram at the right, KJ is tangent to }O. What are the values of x and y?

b. Reasoning In part (a), an inscribed angle (∠Q) and an angle formed by a tangent and chord (∠KJL) intercept the same arc. What is always true of these angles? Explain.

Theorem 12-12

The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.

m∠C = 12 m BDC¬

You will prove Theorem 12-12 in Exercise 34.

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B

C

D

B

C

D

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P

S

R

Q

M

m∠PQS + m∠PQR = 180. So first find m∠PQS using PMQ¬.m∠PQR

• <SR> is tangent to the circle at Q

• m PMQ¬ = 212

The measure of an ∠ formed by a tangent and a chord is 12 the measure of the intercepted arc.

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Q

O

J

L

K

35�

y�x�

Got It?

How can you check the answer? One way is to use m∠PQR to find m PQ¬ . Confirm that m PQ¬ + m PMQ¬ = 360.

784 Chapter 12 Circles

Lesson CheckDo you know HOW?Use the diagram for Exercises 1–3.

1. Which arc does ∠A intercept?

2. Which angle intercepts ABC¬?

3. Which angles of quadrilateral ABCD are supplementary?

Do you UNDERSTAND? 4. Vocabulary What is the relationship between an

inscribed angle and its intercepted arc?

5. Error Analysis A classmate says that m∠A = 90. What is your classmate’s error?

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P

A

D

B

C

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A

B

C

Practice and Problem-Solving Exercises

Find the value of each variable. For each circle, the dot represents the center.

6. 7. 8.

9. 10. 11.

12. 13. 14. 15.

Find the value of each variable. Lines that appear to be tangent are tangent.

16. 17. 18.

19. Writing A parallelogram inscribed in a circle must be what kind of parallelogram? Explain.

PracticeA See Problems 1 and 2.

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108�60�

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See Problem 3.

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MATHEMATICAL PRACTICES

MATHEMATICAL PRACTICES

Lesson 12-3 InscribedAngles 785

Find each indicated measure for O.

20. a. mBC¬ b. m∠B

c. m∠C

d. mAB¬

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C

O

B

A48�

110� 21. a. m∠A

b. m CE¬ c. m∠C

d. m∠D

e. m∠ABE

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B

C

D

O

E

A

80�

80�

25�

22. Think About a Plan What kind of trapezoid can be inscribed in a circle? Justify your response.

• Draw several diagrams to make a conjecture. • How can parallel lines help?

Find the value of each variable. For each circle, the dot represents the center.

23. 24. 25.

Write a proof for Exercises 26 and 27.

26. Inscribed Angle Theorem, Case II

Given: }O with inscribed ∠ABC

Prove: m∠ABC = 12 mAC¬

(Hint: Use the Inscribed Angle Theorem, Case I.)

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Proof

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OA

B

PC

27. Inscribed Angle Theorem, Case III

Given: }S with inscribed ∠PQR

Prove: m∠PQR = 12 m PR¬

(Hint: Use the Inscribed Angle Theorem, Case I.)

Proof

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R

Q

T

PS

28. Television The director of a telecast wants the option of showing the same scene from three different views.

a. Explain why cameras in the positions shown in the diagram will transmit the same scene.

b. Reasoning Will the scenes look the same when the director views them on the control room monitors? Explain.

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Camera 1 Camera

3Camera 2

Scene

786 Chapter 12 Circles

30. Constructions The diagrams below show the construction of a tangent to a circle from a point outside the circle. Explain why

<BC

> must be tangent to }A. (Hint: Copy

the third diagram and draw AC.)

Write a proof for Exercises 31–34.

31. Inscribed Angle Theorem, Corollary 1

Given: }O, ∠A intercepts BC¬, ∠D intercepts BC¬.

Prove: ∠A ≅ ∠D

O O BA B

CC

A BA

Given: �A and point BConstruct the midpointof AB. Label the point O.�

Construct a semicircle withradius OA and center O. Labelits intersection with �A as C.

Draw BC.

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O

Proof

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BD

C

A

O

33. Inscribed Angle Theorem, Corollary 3

Given: Quadrilateral ABCD inscribed in }O

Prove: ∠A and ∠C are supplementary. ∠B and ∠D are supplementary.

Proof

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B

D C

A

O

Reasoning Is the statement true or false? If it is true, give a convincing argument. If it is false, give a counterexample.

35. If two angles inscribed in a circle are congruent, then they intercept the same arc.

36. If an inscribed angle is a right angle, then it is inscribed in a semicircle.

37. A circle can always be circumscribed about a quadrilateral whose opposite angles are supplementary.

ChallengeC

32. Inscribed Angle Theorem, Corollary 2

Given: }O with ∠CAB inscribed in a semicircle

Prove: ∠CAB is a right angle.

Proof

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B

DC

A

O

34. Theorem 12-12

Given: GH and tangent / intersecting }E at H

Prove: m∠GHI = 12 m GFH¬

Proof

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FG

I

H

E

�

29. Reasoning Can a rhombus that is not a square be inscribed in a circle? Justify your answer.

Lesson 12-3 Inscribed Angles 787

38. Prove that if two arcs of a circle are included between parallel chords, then the arcs are congruent.

39. Constructions Draw two segments. Label their lengths x and y. Construct the geometric mean of x and y. (Hint: Recall a theorem about a geometric mean.)

Proof

Apply What You’ve Learned

Look back at the information given on page 761 about the logo for the showroom display. The diagram of the logo is shown again below.

Consider relationships of angles and arcs in the diagram. Select all of the following that are true. Explain your reasoning.

7.2 ft9 ft

15 ft27 ft

DB

G

C

A

O

E

A. ∠ADB is an inscribed angle in }O.

B. ∠AGB is an inscribed angle in }O.

C. ∠AGB intercepts GB¬.

D. ∠AGB intercepts AEB¬.

E. The measure of AG¬ is half the measure of ∠AGB.

F. △AGB is a right triangle.

G. △ABG ∼ △BDG

PERFO

RMANCE TA

SK MATHEMATICAL PRACTICESMP 1