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10.5 Relationships in

EEssentialssential QQuestionuestion When a intersects a or another chord, what relationships exist among the and arcs formed?

Angles Formed by a Chord and Tangent Line with a partner. Use dynamic geometry software. a. Construct a chord in a . At Sample one of the endpoints of the chord, construct a tangent line to the circle.

b. Find the measures of the two angles C B formed by the chord and the tangent line. c. Find the measures of the two A circular arcs determined by the chord. d. Repeat parts (a)–(c) several times. D Record your results in a table. Then write a conjecture that summarizes the .

Angles Formed by Intersecting Chords Work with a partner. Use dynamic geometry software. a. Construct two chords that Sample intersect inside a circle.

b. Find the of one D CONSTRUCTING of the angles formed by the B VIABLE ARGUMENTS intersecting chords. To be profi cient in math, c. Find the measures of the arcs C you need to understand intercepted by the angle in part (b) A and use stated assumptions, and its vertical angle. What do defi nitions, and previously you observe? E established results. d. Repeat parts (a)–(c) several times. Record your results in a table. Then write a conjecture that summarizes the data. CCommunicateommunicate YourYour AnswerAnswer 3. When a chord intersects a tangent line or another chord, what relationships exist 148° among the angles and arcs formed? m m ∠ 1 4. Line is tangent to the circle in the fi gure at the left. Find the measure of 1. 5. Two chords intersect inside a circle to form a pair of vertical angles with measures of 55°. Find the sum of the measures of the arcs intercepted by the two angles.

Section 10.5 Angle Relationships in Circles 561

hhs_geo_pe_1005.indds_geo_pe_1005.indd 561561 11/19/15/19/15 2:372:37 PMPM 10.5 Lesson WWhathat YYouou WWillill LLearnearn Find angle and arc measures. Core VocabularyVocabulary Use circumscribed angles. circumscribed angle, p. 564 Finding Angle and Arc Measures Previous tangent TTheoremheorem chord secant Theorem 10.14 Tangent and Intersected Chord Theorem If a tangent and a chord intersect at a on a circle, B then the measure of each angle formed is one-half the C measure of its intercepted arc. 2 1 A

m∠1 = 1 mAB m∠2 = 1 mBCA Proof Ex. 33, p. 568 2 2

Finding Angle and Arc Measures

Line m is tangent to the circle. Find the measure of the red angle or arc. a. A b. K m

130° 125° L 1 J B m

SOLUTION 1 m∠ = — ° = ° mKJL = ° = ° a. 1 2 (130 ) 65 b. 2(125 ) 250

MMonitoringonitoring ProgressProgress Help in English and Spanish at BigIdeasMath.com Line m is tangent to the circle. Find the indicated measure. 1. m∠1 2. mRST 3. mXY

m m S m 1 ° Y 80° T 98 X 210° R

CCoreore CConceptoncept Intersecting Lines and Circles If two nonparallel lines intersect a circle, there are three places where the lines can intersect.

on the circle inside the circle outside the circle

562 Chapter 10 Circles

hhs_geo_pe_1005.indds_geo_pe_1005.indd 562562 11/19/15/19/15 2:372:37 PMPM TTheoremsheorems Theorem 10.15 Angles Inside the Circle Theorem D If two chords intersect inside a circle, then the measure of A sum 1 each angle is one-half the of the measures of the arcs 2 C intercepted by the angle and its vertical angle. B m∠1 = 1 (mDC + mAB ), 2 Proof Ex. 35, p. 568 m∠2 = 1 (mAD + mBC ) 2

Theorem 10.16 Angles Outside the Circle Theorem If a tangent and a secant, two , or two secants intersect outside a circle, then the measure of the angle formed is one-half the difference of the measures of the intercepted arcs.

B P X Q A W 1 2 3 Z C R Y m∠1 = 1 (mBC − mAC ) m∠2 = 1 (mPQR − mPR ) m∠3 = 1 (mXY − mWZ ) 2 2 2

Proof Ex. 37, p. 568

Finding an Angle Measure

Find the value of x. a. ° b. C D 130 M ° x° B 76 J L x° 178° K A 156°

SOLUTION — — a. The chords JL and KM intersect b. The tangent CD៮ ៮៮⃗ and the secant CB៮ ៮៮⃗ inside the circle. Use the Angles intersect outside the circle. Use the Inside the Circle Theorem. Angles Outside the Circle Theorem.

1 1 x° = — mJM + mLK m∠BCD = — mAD − mBD 2 ( ) 2 ( ) 1 1 x° = — ° + x° = — ° − 2 ( 130 156°) 2 ( 178 76°) x = 143 x = 51 So, the value of x is 143. So, the value of x is 51.

MMonitoringonitoring ProgressProgress Help in English and Spanish at BigIdeasMath.com Find the value of the .

4. y° 5. F A B a° 102° K 44° C G D J H 95° 30°

Section 10.5 Angle Relationships in Circles 563

hhs_geo_pe_1005.indds_geo_pe_1005.indd 563563 11/19/15/19/15 2:372:37 PMPM Using Circumscribed Angles CCoreore CConceptoncept Circumscribed Angle A

A circumscribed angle is an angle B whose sides are tangent to a circle. circumscribed C angle TTheoremheorem Theorem 10.17 Circumscribed Angle Theorem A The measure of a circumscribed angle is equal to 180° minus the measure of the that intercepts D C the same arc. B

Proof Ex. 38, p. 568 m∠ADB = 180° − m∠ACB

Finding Angle Measures

Find the value of x. a. D b. E A J x° x° ° G 135 ° C H 30 B F

SOLUTION a. Use the Circumscribed Angle Theorem to fi nd m∠ADB. m∠ADB = 180° − m∠ACB Circumscribed Angle Theorem x° = 180° − 135° Substitute. x = 45 Subtract. So, the value of x is 45.

b. Use the Measure of an Theorem (Theorem 10.10) and the Circumscribed Angle Theorem to fi nd m∠EJF. 1 m∠EJF = — mEF 2 Measure of an Inscribed Angle Theorem 1 m∠EJF = — m∠EGF 2 Defi nition of minor arc 1 m∠EJF = — ° − m∠EHF 2 (180 ) Circumscribed Angle Theorem 1 m∠EJF = — ° − ° 2 (180 30 ) Substitute. 1 x = — − 2 (180 30) Substitute. x = 75 Simplify. So, the value of x is 75.

564 Chapter 10 Circles

hhs_geo_pe_1005.indds_geo_pe_1005.indd 564564 11/19/15/19/15 2:372:37 PMPM Modeling with Mathematics

The northern lights are bright fl ashes of colored C light between 50 and 200 miles above . A BD fl ash occurs 150 miles above Earth at point C. What is the measure of BD , the portion of Earth from which the fl ash is visible? (Earth’s 4150 mi 4000 mi is approximately 4000 miles.) A

E SOLUTION Not drawn to scale 1. Understand the Problem You are given the approximate radius of Earth and the above Earth that the fl ash occurs. You need to fi nd the measure of the arc that represents the portion of Earth from which the fl ash is visible. 2. Make a Plan Use properties of tangents, , and angles outside a circle to fi nd the arc measure. — — — — — — 3. Solve the Problem Because CB and CD are tangents, CB ⊥ AB and CD ⊥ AD BC— ≅ DC— by the Tangent Line to Circle Theorem (Theorem 10.1). Also, — — by the External Tangent Congruence Theorem (Theorem 10.2), and CA ≅ CA by the Refl exive Property of Congruence (Theorem 2.1). So, △ABC ≅ △ADC by the -Leg Congruence Theorem (Theorem 5.9). Because corresponding COMMON ERROR parts of congruent are congruent, ∠BCA ≅ ∠DCA. Solve right △CBA to fi nd that m∠BCA ≈ 74.5°. So, m∠BCD ≈ 2(74.5°) = 149°. Because the value for m∠BCD is an m∠BCD = 180° − m∠BAD Circumscribed Angle Theorem approximation, use the m∠BCD = 180° − mBD Defi nition of minor arc symbol ≈ instead of =. 149° ≈ 180° − mBD Substitute. 31° ≈ mBD Solve for m BD . The measure of the arc from which the fl ash is visible is about 31°.

4. Look Back You can use inverse trigonometric to fi nd m∠BAC and m∠DAC.

m∠BAC = cos−1 —4000 ≈ 15.5° ( 4150) 4000 m∠DAC = cos−1 — ≈ 15.5° ( 4150) So, m∠BAD ≈ 15.5° + 15.5° = 31°, and therefore mBD ≈ 31°.

Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the value of x.

6. 7. T K x° P B R N x° 120° M

CD Q L 50° 4002.73 mi S A 4000 mi 8. You are on top of Mount Rainier on a clear day. You are about 2.73 miles above sea level at point B. Find mCD , which represents the part of Earth that you can see. Not drawn to scale

Section 10.5 Angle Relationships in Circles 565

hs_geo_pe_1005.indd 565 3/9/16 9:31 AM 10.5 Exercises Dynamic Solutions available at BigIdeasMath.com

VVocabularyocabulary andand CoreCore ConceptConcept CheckCheck 1. COMPLETE THE SENTENCE Points A, B, C, and D are on a circle, and ⃖ AB៮៮⃗ intersects CD⃖ ៮៮⃗ at point P. 1 m∠APC = — mBD − mAC P If 2 ( ), then point is ______the circle.

2. WRITING Explain how to fi nd the measure of a circumscribed angle.

MMonitoringonitoring ProgressProgress andand ModelingModeling withwith MathematicsMathematics

In Exercises 3–6, line t is tangent to the circle. Find the M 13. N 14. 17x° indicated measure. (See Example 1.) x° 73° 3. mAB 4. mDEF L F P B t E H G 75° 117° D 65° ERROR ANALYSIS In Exercises 15 and 16, describe and A t F correct the error in fi nding the angle measure. m∠ m∠ 5. 1 6. 3 15. ✗ Q 37°R ° 140 t U m∠SUT = mST = 46° t P T So, m∠SUT = 46°. ° 260 S 46° 1 3 16. ✗ 122° m∠1 = 122° − 70° In Exercises 7–14, fi nd the value of x. (See Examples 2 70° = 52° and 3.) 1 So, m∠1 = 52°. 7. C 8. K B

x° 145° J x − ° In Exercises 17–22, fi nd the indicated angle measure. 85° (2 30) 30° Justify your answer. M A D x° L 120° 3 120° D ° U 9. 29 E 10. 34° T (x + 6)° x° F 114° S (3x − 2)° W 60° 2 G V 6 4 1 5

11. Q (x + 30)° 12. 17. m∠1 18. m∠2 (x + 70)° R x ° Y ( 2 ) 19. m∠3 20. m∠4 P S 125° m∠ m∠ T X 21. 5 22. 6 (6x − 11)° Z 566 Chapter 10 Circles

hhs_geo_pe_1005.indds_geo_pe_1005.indd 566566 11/19/15/19/15 2:372:37 PMPM ៮៮៮⃗ 23. PROBLEM SOLVING You are fl ying in a hot air 27. ABSTRACT REASONING— In the diagram, PL is tangent balloon about 1.2 miles above the ground. Find the to the circle, and KJ is a . What is the range measure of the arc that represents the part of Earth of possible angle measures of ∠LPJ? Explain your

you can see. The radius of Earth is about 4000 miles. reasoning. (See Example 4.) P L C W K Z X J

4001.2 mi — 4000 mi 28. ABSTRACT REASONING In the diagram, AB is any Y chord that is not a diameter of the circle. Line m is tangent to the circle at point A. What is the range of possible values of x? Explain your reasoning. (The Not drawn to scale diagram is not drawn to scale.)

B m 24. PROBLEM SOLVING You are watching fi reworks over San Diego Bay S as you sail away in a boat. x° The highest point the fi reworks reach F is about A 0.2 mile above the bay. Your eyes E are about 0.01 mile above the water. At point B you can no longer see the fi reworks because of the of ⃖JL៮៮⃗ ⃖NL៮៮⃗ Earth.— The radius of Earth is about 4000 miles, 29. PROOF In the diagram, and are secant lines and FE is tangent to Earth at point T. Find mSB . that intersect at point L. Prove that m∠JPN > m∠JLN. Round your answer to the nearest tenth. F J

S K T P L

E M B N C

Not drawn to scale 30. MAKING AN ARGUMENT Your friend claims that it is 25. MATHEMATICAL CONNECTIONS In the diagram, ៮ BA៮៮⃗ possible for a circumscribed angle to have the same measure as its intercepted arc. Is your friend correct? is tangent to ⊙E. Write an algebraic expression for mCD in terms of x. Then fi nd mCD . Explain your reasoning.

A 31. REASONING Points A and B are on a circle, and t is a 40° tangent line containing A and another point C. x° x° B 7 E 3 C a. Draw two diagrams that illustrate this situation. D b. Write an for mAB in terms of m∠BAC for each diagram. ∠BAC 26. MATHEMATICAL CONNECTIONS The circles in the c. For what measure of can you use either mAB diagram are concentric. Write an algebraic expression equation to fi nd ? Explain. for c in terms of a and b. 32. REASONING △XYZ is an AB— ⊙P X inscribed— in ⊙P. is tangent to — at point , BC is tangent to ⊙P at point Y, and AC is tangent to a° ⊙P at point Z. Draw a diagram that illustrates this b° situation. Then classify △ABC by its angles and sides. c° Justify your answer.

Section 10.5 Angle Relationships in Circles 567

hhs_geo_pe_1005.indds_geo_pe_1005.indd 567567 11/19/15/19/15 2:372:37 PMPM 33. PROVING A THEOREM To prove the Tangent and 36. THOUGHT PROVOKING In the fi gure, ⃖ BP៮៮⃗ and CP⃖ ៮៮⃗ are Intersected Chord Theorem (Theorem 10.14), you tangent to the circle. Point A is any point —on the major must prove three cases. arc formed by the endpoints of the chord BC . Label all — congruent angles in the fi gure. Justify your reasoning. a. The diagram shows the case where AB contains the center of the circle. Use the Tangent Line C to Circle Theorem (Theorem 10.1) to write a paragraph proof for this case. A B P

B

C A 37. PROVING A THEOREM Use the diagram below b. Draw a diagram and write a proof for the case to prove the Angles Outside the Circle Theorem where the center of the circle is in the interior (Theorem 10.16) for the case of a tangent and a of ∠CAB. secant. Then copy the diagrams for the other two c. Draw a diagram and write a proof for the case cases on page 563 and draw appropriate auxiliary where the center of the circle is in the exterior segments. Use your diagrams to prove each case. ∠CAB of . B A 34. HOW DO YOU SEE IT? In the diagram, television 1 2 cameras are positioned at A and B to record what C happens on stage. The stage is an arc of ⊙A. You would like the camera at B to have a 30° view of the 38. PROVING A THEOREM Prove that the Circumscribed stage. Should you move the camera closer or farther Angle Theorem (Theorem 10.17) follows from the away? Explain your reasoning. Angles Outside the Circle Theorem (Theorem 10.16).

In Exercises 39 and 40, fi nd the indicated measure(s). 80° Justify your answer. 39. Find m∠P when mWZY = 200°. A W X 30° P Z Y 25° B 40. Find mAB and mED .

60° A 20° J E F G

H 35. PROVING A THEOREM Write a proof of the Angles 115° B Inside the Circle Theorem (Theorem 10.15). — — Given Chords AC and BD D intersect inside a circle. A D 1 m∠ = 1 mDC + mAB C Prove 1 — ( ) C ° 2 B 85

MMaintainingaintaining MathematicalMathematical ProficiencyProficiency Reviewing what you learned in previous grades and lessons Solve the equation. (Skills Review Handbook) 41. x2 + x = 12 42. x2 = 12x + 35 43. −3 = x2 + 4x

568 Chapter 10 Circles

hhs_geo_pe_1005.indds_geo_pe_1005.indd 568568 11/19/15/19/15 2:372:37 PMPM