Big Ideas Chapter 10 Section 5 Lecture Notes Practice Questions

10.5 Angle Relationships in Circles Essential Question When a chord intersects a tangent line or another chord, what relationships exist among the angles and arcs formed? Angles Formed by a Chord and Tangent Line Work with a partner. Use dynamic geometry software. a. Construct a chord in a circle. At Sample one of the endpoints of the chord, construct a tangent line to the circle. b. Find the measures of the two angles formed by the chord and the tangent line. C B c. Find the measures of the two circular arcs determined by the chord. A D d. Repeat parts (a)–(c) several times. Record your results in a table. Then write a conjecture that summarizes the data. Angles Formed by Intersecting Chords Work with a partner. Use dynamic geometry software. a. Construct two chords that intersect inside a circle. CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, you need to understand and use stated assumptions, definitions, and previously established results. Sample b. Find the measure of one of the angles formed by the intersecting chords. D B c. Find the measures of the arcs intercepted by the angle in part (b) and its vertical angle. What do you observe? C A E d. Repeat parts (a)–(c) several times. Record your results in a table. Then write a conjecture that summarizes the data. Communicate You Communicate Your Your our Answer Answer 148° 1 m 3. When a chord intersects a tangent line or another chord, what relationships exist among the angles and arcs formed? 4. Line m is tangent to the circle in the figure 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 10.5 Lesson What You Learn arn You Will Will Lear Le arn Find angle and arc measures. Use circumscribed angles. Core Vocabular Vocabulary ry circumscribed angle, p. 564 Finding Angle and Arc Measures Previous tangent chord secant Theorem Theorem Theorem 10.14 Tangent and Intersected Chord Theorem If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one-half the measure of its intercepted arc. B C 2 1 A m∠1 = 12 mAB Proof Ex. 33, p. 568 m∠2 = 1mBCA 2 Finding Angle and Arc Measures Line m is tangent to the circle. Find the measure of the red angle or arc. a. b. A 130° 1 B K J m m 125° L SOLUTION b. m KJL = 2(125°) = 250° a. m∠1 = 1—2 (130°) = 65° Monitoring Progress Help in English and Spanish at BigIdeasMath.com Line m is tangent to the circle. Find the indicated measure. 2. m RST 1. m∠1 1 m 210° m T XY 3. m m S Y 98° 80° R X Core Concept Core Intersecting Lines and Circles If two nonparallel lines intersect a circle, there are three places where the lines canintersect. on the circle 562 Chapter 10 Circles inside the circle outside the circle Theor Theorem Theorems ems ems Theorem 10.15 Angles Inside the Circle Theorem If two chords intersect inside a circle, then the measure of each angle is one-half the sum of the measures of the arcs intercepted by the angle and itsvertical angle. D A 1 2 C B m∠1 = 12 (mDC + mAB ), Proof Ex. 35, p. 568 1 m∠2 = 2 (mAD + mBC ) Theorem 10.16 Angles Outside the Circle Theorem If a tangent and a secant, two tangents, 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 3 Z 2 1 C R m∠1 = 1 (mBC − mAC ) m∠2 = 1 (mPQR − mPR ) 2 Y m∠3 = 12 (mXY − mWZ ) 2 Proof Ex. 37, p. 568 Finding an Angle Measure Find the value of x. a. 130° J b. C M x° L x° D B 76° 178° K A 156° SOLUTION a. The chords — JL and — KM intersect insidethe circle. Use the Angles Insidethe Circle Theorem. b. The tangent  CD and the secant  CB intersect outside the circle. Use the Angles Outside the Circle Theorem. LK ) x° = 1—2 (m JM + m  x° = 1 —(130° 2 BD ) m∠BCD = 1—2 (m AD − m x° = 12—(178° − 76°) + 156°) x = 143 x = 51 So, the value of x is 143. Monitoring Progress So, the value of x is 51. Help in English and Spanish at BigIdeasMath.com Find the value of the variable. 4. A y° 5. F B 102° K C D J 95° Section 10.5 a° 44° H 30° G Angle Relationships in Circles 563 Using Circumscribed Angles Core Concept Core Circumscribed Angle A A circumscribed angle is an angle whose sides are tangent to a circle. B circumscribed C angle Theorem Theorem Theorem 10.17 Circumscribed Angle Theorem The measure of a circumscribed angle is equal to 180° minus the measure of the central angle that intercepts the same arc. A D C B Proof Ex. 38, p. 568 m∠ ADB = 180° − m∠ ACB Finding Angle Measures Find the value of x. a. D A b. E J x° 135° x° H C 30° B G F SOLUTION a. Use the Circumscribed Angle Theorem to find m∠ADB. m∠ADB = 180° − m∠ACB x° = 180° − 135° x = 45 Circumscribed Angle Theorem Substitute. Subtract. So, the value of x is 45. b. Use the Measure of an Inscribed Angle Theorem (Theorem 10.10) and the Circumscribed Angle Theorem to find m∠EJF. m∠EJF = —12m EF Measure of an Inscribed Angle Theorem m∠EJF = —12m∠EGF Definition of minor arc m∠EJF = —12(180° − m∠EHF) Circumscribed Angle Theorem m∠EJF = —12(180° − 30°) Substitute. x = —12(180 − 30) Substitute. x = 75 Simplify. So, the value of x is 75. 564 Chapter 10 Circles Modeling with Mathematics The northern lights are bright flashes of colored light between 50 and 200 miles above Earth. A B flash occurs 150 miles above Earth at point C. What is the measure of  BD , the portion of Earth from which the flash is visible? (Earth’s radius 4150 mi is approximately 4000 miles.) C D 4000 mi A E SOLUTION Not drawn to scale 1. Understand the Problem You are given the approximate radius of Earth and the distance above Earth that the flash occurs. You need to find the measure of the arc that represents the portion of Earth from which the flash is visible. 2. Make a Plan Use properties of tangents, triangle congruence, and angles outside acircle to find the arc measure. — are tangents, CB — ⊥ AB — and CD — ⊥ AD — 3. Solve the Problem Because — CB andCD —≅ DC — by by the Tangent Line to Circle Theorem (Theorem 10.1). Also, BC — — the External Tangent Congruence Theorem (Theorem 10.2), and CA ≅ CA by the Reflexive Property of Congruence (Theorem 2.1). So, △ABC ≅ △ADC by the Hypotenuse-Leg Congruence Theorem (Theorem 5.9). Because corresponding parts of congruent triangles are congruent, ∠BCA ≅ ∠DCA. Solve right △CBA to find that m∠BCA≈ 74.5°. So, m∠BCD ≈ 2(74.5°) = 149°. COMMON ERROR Because the value for m∠BCD is an approximation, use the symbol ≈ instead of =. m∠BCD = 180° − m∠BAD Circumscribed Angle Theorem BD m∠BCD = 180° − m Definition of minor arc BD 149° ≈ 180° − m Substitute. BD 31° ≈ m . Solve for mBD The measure of the arc from which the flash is visible is about 31°. 4. Look Back You can use inverse trigonometric ratios to find m∠BAC and m∠DAC. ( ) 4000 ( 4150 ) ≈ 15.5° 4000 m∠BAC = cos−1 — ≈ 15.5° 4150 m∠DAC = cos−1 — So, m∠BAD ≈ 15.5° + 15.5° = 31°, and therefore m BD ≈ 31°. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the value of x. 6. 7. T K x° P B N C x° R 120° M D Q L 4002.73 mi A 50° S 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 m CD , which represents the part of Earth that you can see. Not drawn to scale Section 10.5 Angle Relationships in Circles 565 10.5 Exercises Dynamic Solutions available at BigIdeasMath.com VVocabulary ocabulary ocabulary and Core Concept Concept Check  intersects  CD at point P. 1. COMPLETE THE SENTENCE Points A, B, C, and D are on a circle, and AB AC ), then point P is ______ the circle. Ifm∠APC = 1—2 (m  BD − m 2. WRITING Explain how to find the measure of a circumscribed angle. Monitoring Progress Progress and Modeling Modeling with Mathematics Mathematics In Exercises 3– 6, line t is tangent to the circle. Find the indicated measure. (See Example 1.) 3. m AB 13. M 4. m DEF 14. N x° L 73° 17x° F P B t E G H 117° D 65° ERROR ANALYSIS In Exercises 15 and 16, describe and A t F 5. m∠1 correct the error in finding the angle measure. 15. 6. m∠3 140° t ✗ Q 37° R U t P T S 46° 260° 1 3 16. ✗ C 8. m∠SUT = m ST = 46° So, m∠SUT = 46°. 122° m∠1 = 122° − 70° = 52° So, m∠1 = 52°. 70° In Exercises 7–14, find the value of x. (See Examples 2 and 3.) 7. 75° 1 K B 145° x° 85° A J 30° M D (2x − 30)° x° In Exercises 17–22, find the indicated angle measure. Justify your answer. L 120° D 29° 9. x° 114° 10. E U 34° T F (3x − 2)° S W Q 12. (x + 30)° R ( x 2 4 2 1 18. m∠2 19. m∠3 20. m∠4 21. m∠5 22. m∠6 125° S Chapter 10 17. m∠1 Y )° T 566 60° 5 (x + 70)° P 6 V G 11. 120° 3 (x + 6)° Circles X (6x − 11)° Z 23. PROBLEM SOLVING You are flying in a hot air balloon about 1.2 miles above the ground. Find the measure of the arc that represents the part of Earth you can see. The radius of Earth is about 4000 miles. (See Example 4.) C 27. ABSTRACT REASONING In the diagram,  PL is tangent to the circle, and — KJ is a diameter. What is the range of possible angle measures of ∠LPJ? Explain your reasoning. L P W K Z X J 4001.2 mi 4000 mi Y Not drawn to scale AB is any 28. ABSTRACT REASONING In the diagram, — 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 diagram is not drawn to scale.) B m 24. PROBLEM SOLVING You are watching fireworks over San Diego Bay S as you sail away in a boat. The highest point the fireworks reach F is about 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 fireworks because of the curvature of Earth. The radius of Earth is about 4000 miles, — is tangent to Earth at point T. Find m  and FE SB . Round your answer to the nearest tenth. x° A 29. PROOF In the diagram,  JL and  NL are secant lines that intersect at point L. Prove that m∠JPN > m∠JLN. F J S K T L P M E B N C Not drawn to scale 30. MAKING AN ARGUMENT Your friend claims that it is 25. MATHEMATICAL CONNECTIONS In the diagram,  BA is tangent to ⊙E. Write an algebraic expression for m CD in terms of x. Then find m CD . A 7x° E 3x° possible for a circumscribed angle to have the same measure as its intercepted arc. Is your friend correct? Explain your reasoning. 31. REASONING Points A and B are on a circle, and t is a 40° B C tangent line containing A and another point C. a. Draw two diagrams that illustrate this situation. b. Write an equation for m  AB in terms of m∠BAC for each diagram. D 26. MATHEMATICAL CONNECTIONS The circles in the diagram are concentric. Write an algebraic expression for c in terms of a and b. a° b° c° c. For what measure of ∠BAC can you use either equation to find m AB ? Explain. 32. REASONING △XYZ is an equilateral triangle — is tangent to ⊙P at point X, inscribed in ⊙P. AB — istangent to ⊙P at point Y, and AC — is tangent to BC ⊙P at point Z. Draw a diagram that illustrates this situation. Then classify △ABC by its angles and sides. Justify your answer. Section 10.5 Angle Relationships in Circles 567 33. PROVING A THEOREM To prove the Tangent and Intersected Chord Theorem (Theorem 10.14), you must prove three cases. a. — The diagram shows the case where AB contains the center of the circle. Use the Tangent Line to Circle Theorem (Theorem 10.1) to write a paragraph proof for this case.   and  CP are 36. THOUGHT PROVOKING In the figure, BP tangent to the circle. Point A is any point on the major arc formed by the endpoints of the chord — BC . Label all congruent angles in the figure. Justify your reasoning. C A B P B C A b. Draw a diagram and write a proof for the case where the center of the circle is in the interior of ∠CAB. c. Draw a diagram and write a proof for the case where the center of the circle is in the exterior of∠CAB. 37. PROVING A THEOREM Use the diagram below to prove the Angles Outside the Circle Theorem (Theorem 10.16) for the case of a tangent and a secant. Then copy the diagrams for the other two cases on page 563 and draw appropriate auxiliary segments. Use your diagrams to prove each case. B A 1 34. HOW DO YOU SEE IT? In the diagram, television cameras are positioned at A and B to record what happens on stage. The stage is an arc of ⊙A. You would like the camera at B to have a 30° view of the stage. Should you move the camera closer or farther away? Explain your reasoning. 2 C 38. PROVING A THEOREM Prove that the Circumscribed Angle Theorem (Theorem 10.17) follows from the Angles Outside the Circle Theorem (Theorem 10.16). In Exercises 39 and 40, find the indicated measure(s). Justify your answer. 80° WZY = 200°. 39. Find m∠P when m  A W X Z Y 30° P 25° 40. Find m AB and m  ED . B 60° A 20° J G E F 35. PROVING A THEOREM Write a proof of the Angles Inside the Circle Theorem (Theorem 10.15). Given AC and — BD Chords — intersect inside a circle. AB ) Prove m∠1 = —12(m  DC + m 115° H D A D 1 B C Maintaining Mathematical Proficiency iciency Mathematical Proficienc Prof iciency 85° 568 Chapter 10 42. x2 = 12x + 35 Circles C Reviewing what you learned in previous grades and lessons Solve the equation. (Skills Review Handbook) 41. x2 + x = 12 B 43. −3 = x2 + 4x