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10.5 Angle Relationships in CirclesEssential QuestionWhen a chord intersects a tangent line oranother chord, what relationships exist among the angles and arcs formed?Angles Formed by a Chord and Tangent LineWork with a partner. Use dynamic geometry software.a. Construct a chord in a circle. AtSampleone of the endpoints of the chord,construct a tangent line to the circle.b. Find the measures of the two anglesformed by the chord and thetangent line.CBAc. Find the measures of the twocircular arcs determined bythe chord.Dd. Repeat parts (a)–(c) several times.Record your results in a table.Then write a conjecture thatsummarizes the data.Angles Formed by Intersecting ChordsWork with a partner. Use dynamic geometry software.a. Construct two chords thatintersect inside a circle.CONSTRUCTINGVIABLE ARGUMENTSTo be proficient in math,you need to understandand use stated assumptions,definitions, and previouslyestablished results.Sampleb. Find the measure of oneof the angles formed by theintersecting chords.DBc. Find the measures of the arcsintercepted by the angle in part (b)and its vertical angle. What doyou observe?CAEd. Repeat parts (a)–(c) several times.Record your results in a table.Then write a conjecture thatsummarizes the data.Communicate Your Answer148 1m3. When a chord intersects a tangent line or another chord, what relationships existamong 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 measuresof 55 . Find the sum of the measures of the arcs intercepted by the two angles.Section 10.5hs geo pe 1005.indd 561Angle Relationships in Circles5611/19/15 2:37 PM
10.5 LessonWhat You Will LearnFind angle and arc measures.Use circumscribed angles.Core VocabulVocabularylarrycircumscribed angle, p. 564Finding Angle and Arc MeasuresPrevioustangentchordsecantTheoremTheorem 10.14 Tangent and Intersected Chord TheoremIf a tangent and a chord intersect at a point on a circle,then the measure of each angle formed is one-half themeasure of its intercepted arc.BC2 1Am 1 12 mABProof Ex. 33, p. 568m 2 12 mBCAFinding Angle and Arc MeasuresLine m is tangent to the circle. Find the measure of the red angle or arc.a.b.A130 1BKJmSOLUTIONm125 Lb. m KJL 2(125 ) 250 a. m 1 —12 (130 ) 65 Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comLine m is tangent to the circle. Find the indicated measure.2. m RST1. m 11m210 mT3. m XYmSY98 80 RXCore ConceptIntersecting Lines and CirclesIf two nonparallel lines intersect a circle, there are three places where the linescan intersect.on the circle562Chapter 10hs geo pe 1005.indd 562inside the circleoutside the circleCircles1/19/15 2:37 PM
TheoremsTheorem 10.15 Angles Inside the Circle TheoremIf two chords intersect inside a circle, then the measure ofeach angle is one-half the sum of the measures of the arcsintercepted by the angle and its vertical angle.DAB21Cm 1 12 (mDC mAB ),Proof Ex. 35, p. 568m 2 12 (mAD mBC )Theorem 10.16 Angles Outside the Circle TheoremIf 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 ofthe intercepted arcs.BPW3Z21m 1 XQACR1(mBC21(mPQR2 mAC )m 2 m 3 mPR )1(mXY2Y mWZ )Proof Ex. 37, p. 568Finding an Angle MeasureFind the value of x.a.130 Jb. CMx B 76 Lx DK178 A156 SOLUTION— and KM— intersecta. The chords JLinside the circle. Use the AnglesInside the Circle Theorem. ⃗ and the secant ⃗b. The tangent CDCBintersect outside the circle. Use theAngles Outside the Circle Theorem. m m BCD —12(mADBD )x —12(m JM m LK )x —12(130 156 )x —12(178 76 )x 143x 51So, the value of x is 143.Monitoring ProgressSo, the value of x is 51.Help in English and Spanish at BigIdeasMath.comFind the value of the variable.4.Ay 5.B102 KCDJ95 Section 10.5hs geo pe 1005.indd 563F30 a 44 HGAngle Relationships in Circles5631/19/15 2:37 PM
Using Circumscribed AnglesCore ConceptCircumscribed AngleAA circumscribed angle is an anglewhose sides are tangent to a circle.Bcircumscribed CangleTheoremTheorem 10.17 Circumscribed Angle TheoremThe measure of a circumscribed angle is equal to 180 minus the measure of the central angle that interceptsthe same arc.ADCBProof Ex. 38, p. 568m ADB 180 m ACBFinding Angle MeasuresFind the value of x.a.DAb.Ex 135 x HC30 BJGFSOLUTIONa. Use the Circumscribed Angle Theorem to find m ADB.m ADB 180 m ACBx 180 135 x 45Circumscribed Angle TheoremSubstitute.Subtract.So, the value of x is 45.b. Use the Measure of an Inscribed Angle Theorem (Theorem 10.10) and theCircumscribed Angle Theorem to find m EJF.EFm EJF —12 m Measure of an Inscribed Angle Theoremm EJF —12 m EGFDefinition of minor arcm EJF —12 (180 m EHF)Circumscribed Angle Theoremm EJF —12 (180 30 )Substitute.x —12(180 30)Substitute.x 75Simplify.So, the value of x is 75.564Chapter 10hs geo pe 1005.indd 564Circles1/19/15 2:37 PM
Modeling with MathematicsThe northern lights are bright flashes of coloredlight between 50 and 200 miles above Earth. ABflash occurs 150 miles above Earth at point C. What is the measure of BD , the portion of Earthfrom which the flash is visible? (Earth’s radius 4150 miis approximately 4000 miles.)CD4000 miAESOLUTIONNot drawn to scale1. Understand the Problem You are given the approximate radius of Earth and thedistance above Earth that the flash occurs. You need to find the measure of the arcthat represents the portion of Earth from which the flash is visible.2. Make a Plan Use properties of tangents, triangle congruence, and angles outsidea circle to find the arc measure.— and CD— are tangents, CB— AB— and CD— AD—3. Solve the Problem Because CB——by the Tangent Line to Circle Theorem (Theorem 10.1). Also, BC DC by— CA— by thethe External Tangent Congruence Theorem (Theorem 10.2), and CAReflexive Property of Congruence (Theorem 2.1). So, ABC ADC by theHypotenuse-Leg Congruence Theorem (Theorem 5.9). Because correspondingparts of congruent triangles are congruent, BCA DCA. Solve right CBAto find that m BCA 74.5 . So, m BCD 2(74.5 ) 149 .COMMON ERRORBecause the valuefor m BCD is anapproximation, use thesymbol instead of .m BCD 180 m BADCircumscribed Angle Theoremm BCD 180 m BDDefinition of minor arc149 180 m BDSubstitute.31 m BD .Solve for m BDThe 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 15.5 m BAC cos 1 —4150m DAC cos 1 —So, m BAD 15.5 15.5 31 , and therefore m BD 31 .Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comFind the value of x.6.7.TKPBCNx R120 MDL4002.73 miAx 50 QS4000 mi8. You are on top of Mount Rainier on a clear day. You are about 2.73 miles above sealevel at point B. Find m CD , which represents the part of Earth that you can see.Not drawn to scaleSection 10.5hs geo pe 1005.indd 565Angle Relationships in Circles5653/9/16 9:31 AM
10.5 ExercisesDynamic Solutions available at BigIdeasMath.comVocabulary and Core Concept Check1. COMPLETE THE SENTENCE Points A, B, C, and D are on a circle, and ⃖ ⃗AB intersects ⃖ ⃗CD at point P.If m APC —12 (m BD m AC ), then point P is the circle.2. WRITING Explain how to find the measure of a circumscribed angle.Monitoring Progress and Modeling with MathematicsIn Exercises 3–6, line t is tangent to the circle. Find theindicated measure. (See Example 1.)AB3. m 13.4. m DEFBLt15.140 t316.In Exercises 7–14, find the value of x. (See Examples 2and 3.)C8.145 29 Dx 10.EF(2x 30) x U34 T122 m 1 122 70 52 So, m 1 52 .70 1In Exercises 17–22, find the indicated angle measure.Justify your answer.120 3(x 6) (3x 2) S(x 30) QR6V12.Y( ) x260 421TChapter 1017. m 118. m 219. m 320. m 421. m 522. m 6125 Shs geo pe 1005.indd 566TS 46 m SUT m ST 46 So, m SUT 46 .5(x 70) 566 U120 WPPLG11.HKJ30 MD114 Q 37 R260 1A75 ERROR ANALYSIS In Exercises 15 and 16, describe andcorrect the error in finding the angle measure.Ft9.FG6. m 3x 17x 117 D5. m 185 73 tEAB14.Nx P65 7.MX(6x 11) ZCircles1/19/15 2:37 PM
23. PROBLEM SOLVING You are flying in a hot airballoon about 1.2 miles above the ground. Find themeasure of the arc that represents the part of Earthyou can see. The radius of Earth is about 4000 miles.(See Example 4.)C27. ABSTRACT REASONING In the diagram, ⃗PL is tangent— is a diameter. What is the rangeto the circle, and KJof possible angle measures of LPJ? Explain yourreasoning.LPWKZXJ4001.2 mi4000 miYNot drawn to scale—28. ABSTRACT REASONING In the diagram, AB is anychord that is not a diameter of the circle. Line m istangent to the circle at point A. What is the range ofpossible values of x? Explain your reasoning. (Thediagram is not drawn to scale.)B24. PROBLEM SOLVING You are watching fireworksover San Diego Bay S as you sail away in a boat.The highest point the fireworks reach F is about0.2 mile above the bay. Your eyes E are about0.01 mile above the water. At point B you can nolonger see the fireworks because of the curvature ofEarth. The radius of Earth is about 4000 miles,— is tangent to Earth at point T. Find m and FESB .Round your answer to the nearest tenth.mx A29. PROOF In the diagram, ⃖ ⃗JL and ⃖ ⃗NL are secant linesthat intersect at point L. Prove that m JPN m JLN.FJSKTMEBNCNot drawn to scale30. MAKING AN ARGUMENT Your friend claims that it is25. MATHEMATICAL CONNECTIONS In the diagram, ⃗BAis tangent to E. Write an algebraic expression form CD in terms of x. Then find m CD .A7x E3x possible for a circumscribed angle to have the samemeasure as its intercepted arc. Is your friend correct?Explain your reasoning.31. REASONING Points A and B are on a circle, and t is a40 BCtangent line containing A and another point C.a. Draw two diagrams that illustrate this situation.AB in terms of m BACb. Write an equation for m for each diagram.D26. MATHEMATICAL CONNECTIONS The circles in thediagram are concentric. Write an algebraic expressionfor c in terms of a and b.a b c c. For what measure of BAC can you use eitherequation 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 toBC is tangent to P at point Y, and AC P at point Z. Draw a diagram that illustrates thissituation. Then classify ABC by its angles and sides.Justify your answer.Section 10.5hs geo pe 1005.indd 567LPAngle Relationships in Circles5671/19/15 2:37 PM
33. PROVING A THEOREM To prove the Tangent andIntersected Chord Theorem (Theorem 10.14), youmust prove three cases.— containsa. The diagram shows the case where ABthe center of the circle. Use the Tangent Lineto Circle Theorem (Theorem 10.1) to write aparagraph proof for this case.36. THOUGHT PROVOKING In the figure, ⃖ ⃗BP and ⃖ ⃗CP aretangent to the circle. Point A is any point on the major—. Label allarc formed by the endpoints of the chord BCcongruent angles in the figure. Justify your reasoning.CABPBACb. Draw a diagram and write a proof for the casewhere the center of the circle is in the interiorof CAB.c. Draw a diagram and write a proof for the casewhere the center of the circle is in the exteriorof CAB.37. PROVING A THEOREM Use the diagram belowto prove the Angles Outside the Circle Theorem(Theorem 10.16) for the case of a tangent and asecant. Then copy the diagrams for the other twocases on page 563 and draw appropriate auxiliarysegments. Use your diagrams to prove each case.BA134. HOW DO YOU SEE IT? In the diagram, televisioncameras are positioned at A and B to record whathappens on stage. The stage is an arc of A. Youwould like the camera at B to have a 30 view of thestage. Should you move the camera closer or fartheraway? Explain your reasoning.2C38. PROVING A THEOREM Prove that the CircumscribedAngle Theorem (Theorem 10.17) follows from theAngles Outside the Circle Theorem (Theorem 10.16).In Exercises 39 and 40, find the indicated measure(s).Justify your answer.80 39. Find m P when m WZY 200 .AWXZY30 P25 40. Find m AB and m ED .B60 A20 JGE F35. PROVING A THEOREM Write a proof of the AnglesInside the Circle Theorem (Theorem 10.15).— and BD—Given Chords ACintersect inside a circle.Prove m 1 m AB )1—2 (mDC115 HBDAD1BCMaintaining Mathematical Proficiency85 CReviewing what you learned in previous grades and lessonsSolve the equation. (Skills Review Handbook)41. x2 x 12568Chapter 10hs geo pe 1005.indd 56842. x2 12x 3543. 3 x2 4xCircles1/19/15 2:37 PM
Proof Ex. 33, p. 568 CCore ore 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 A B C 2 1 m 1 1mAB 2 m 2 1mBCA 2 hhs_geo_pe_1005.indd 562s_geo_pe_1005.indd 562 11/19/15 2:37 PM/19/15 2:37 PM
