0.5 NI NOW N I.5... ngle Relationships in ircles ssential Question When a chord intersects a tangent line or another chord, what relationships exist aong the angles and arcs fored? ngles ored by a hord and angent ine Work with a partner. Use dynaic geoetry software. a. onstruct a chord in a circle. t one of the endpoints of the chord, construct a tangent line to the circle. aple b. ind the easures of the two angles fored by the chord and the tangent line. c. ind the easures of the two circular arcs deterined by the chord. d. Repeat parts (a) (c) several ties. Record your results in a table. hen write a conjecture that suarizes the data. ngles ored by Intersecting hords Work with a partner. Use dynaic geoetry software. IN HI RUN o be proficient in ath, you need to understand and use stated assuptions, definitions, and previously established results. a. onstruct two chords that intersect inside a circle. b. ind the easure of one of the angles fored by the intersecting chords. c. ind the easures of the arcs intercepted by the angle in part (b) and its vertical angle. What do you observe? d. Repeat parts (a) (c) several ties. Record your results in a table. hen write a conjecture that suarizes the data. aple ounicate Your nswer 48 3. When a chord intersects a tangent line or another chord, what relationships exist aong the angles and arcs fored? 4. ine is tangent to the circle in the figure at the left. ind the easure of. 5. wo chords intersect inside a circle to for a pair of vertical angles with easures of 55. ind the su of the easures of the arcs intercepted by the two angles. ection 0.5 ngle Relationships in ircles 565
0.5 esson What You Will earn ore Vocabulary circuscribed angle, p. 568 revious tangent chord secant ind angle and arc easures. Use circuscribed angles. inding ngle and rc easures heore heore 0.4 angent and Intersected hord heore If a tangent and a chord intersect at a point on a circle, then the easure of each angle fored is one-half the easure of its intercepted arc. roof x. 33, p. 57 = = inding ngle and rc easures ine is tangent to the circle. ind the easure of the red angle or arc. a. b. 30 5 OUION a. = (30 ) = 65 b. = (5 ) = 50 onitoring rogress Help in nglish and panish at igideasath.co ine is tangent to the circle. ind the indicated easure... R 3. Y 0 98 R Y 80 ore oncept Intersecting ines and ircles 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 566 hapter 0 ircles
heores heore 0.5 ngles Inside the ircle heore If two chords intersect inside a circle, then the easure of each angle is one-half the su of the easures of the arcs intercepted by the angle and its vertical angle. roof x. 35, p. 57 heore 0.6 ngles Outside the ircle heore If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the easure of the angle fored is one-half the difference of the easures of the intercepted arcs. = ( + ), = ( + ) = ( ) roof x. 37, p. 57 R Q = (QR R ) W 3 Z Y 3 = (Y WZ ) ind the value of x. a. 30 56 inding an ngle easure b. 76 78 OUION a. he chords and intersect inside the circle. Use the ngles Inside the ircle heore. = ( + ) = ( 30 + 56 ) x = 43 o, the value of x is 43. onitoring rogress ind the value of the variable. b. he tangent and the secant intersect outside the circle. Use the ngles Outside the ircle heore. = ( ) = ( 78 76 ) x = 5 o, the value of x is 5. Help in nglish and panish at igideasath.co 4. y 95 0 5. 44 H 30 a ection 0.5 ngle Relationships in ircles 567
Using ircuscribed ngles ore oncept ircuscribed ngle circuscribed angle is an angle whose sides are tangent to a circle. circuscribed angle heore heore 0.7 ircuscribed ngle heore he easure of a circuscribed angle is equal to 80 inus the easure of the central angle that intercepts the sae arc. roof x. 38, p. 57 = 80 inding ngle easures ind the value of x. a. 35 b. H 30 OUION a. Use the ircuscribed ngle heore to find. = 80 ircuscribed ngle heore = 80 35 ubstitute. x = 45 ubtract. o, the value of x is 45. b. Use the easure of an Inscribed ngle heore (heore 0.0) and the ircuscribed ngle heore to find. = easure of an Inscribed ngle heore = efinition of inor arc = (80 H) ircuscribed ngle heore = (80 30 ) ubstitute. x = (80 30) ubstitute. x = 75 o, the value of x is 75. iplify. 568 hapter 0 ircles
odeling with atheatics he northern lights are bright flashes of colored light between 50 and 00 iles above arth. flash occurs 50 iles above arth at point. What is the easure of, the portion of arth fro which the flash is visible? (arth s radius is approxiately 4000 iles.) 450 i 4000 i OON RROR ecause the value for is an approxiation, use the sybol instead of =. OUION. Understand the roble You are given the approxiate radius of arth and the distance above arth that the flash occurs. You need to find the easure of the arc that represents the portion of arth fro which the flash is visible.. ake a lan Use properties of tangents, triangle congruence, and angles outside a circle to find the arc easure. 3. olve the roble ecause and are tangents, and by the angent ine to ircle heore (heore 0.). lso, by the xternal angent ongruence heore (heore 0.), and by the Reflexive roperty of ongruence (heore.). o, by the Hypotenuse-eg ongruence heore (heore 5.9). ecause corresponding parts of congruent triangles are congruent,. olve right to find that 74.5. o, (74.5 ) = 49. = 80 = 80 49 80 3 ircuscribed ngle heore efinition of inor arc ubstitute. olve for. he easure of the arc fro which the flash is visible is about 3. Not drawn to scale 4. ook ack You can use inverse trigonoetric ratios to find and. = cos ( 450) 4000 5.5 = cos ( 450) 4000 5.5 o, 5.5 + 5.5 = 3, and therefore 3. onitoring rogress ind the value of x. Help in nglish and panish at igideasath.co 400.73 i Not drawn to scale 4000 i 6. N 0 8. You are on top of ount Rainier on a clear day. You are about.73 iles above sea level at point. ind, which represents the part of arth that you can see. 7. 50 R Q ection 0.5 ngle Relationships in ircles 569
0.5 xercises ynaic olutions available at igideasath.co Vocabulary and ore oncept heck. O H NN oints,,, and are on a circle, and If = ( ), then point is the circle.. WRIIN xplain how to find the easure of a circuscribed angle. onitoring rogress and odeling with atheatics intersects at point. In xercises 3 6, line t is tangent to the circle. ind the indicated easure. (ee xaple.) 3. 65 t 4. 5. 6. 3 t 60 t 7 40 In xercises 7 4, find the value of x. (ee xaples and 3.) 7. 9. 85 4 45 8. 30 3 t (x 30) 9 0. U 34 (x + 6) (3x ) W V. Q (x + 30). (x + 70) R x ( ) Y 5 (6x ) Z 3. 73 N 4. H 75 7 RROR NYI In xercises 5 and 6, describe and correct the error in finding the angle easure. 5. 6. Q 37 R U 46 70 U = = 46 o, U = 46. = 70 = 5 o, = 5. In xercises 7, find the indicated angle easure. ustify your answer. 0 6 3 60 4 5 0 7. 8. 9. 3 0. 4. 5. 6 570 hapter 0 ircles
3. RO OVIN You are flying in a hot air balloon about. iles above the ground. ind the easure of the arc that represents the part of arth you can see. he radius of arth is about 4000 iles. (ee xaple 4.) Z W 7. R RONIN In the diagra, is tangent to the circle, and is a diaeter. What is the range of possible angle easures of? xplain your reasoning. 400. i Y Not drawn to scale 4000 i 8. R RONIN In the diagra, is any chord that is not a diaeter of the circle. ine is tangent to the circle at point. What is the range of possible values of x? xplain your reasoning. (he diagra is not drawn to scale.) 4. RO OVIN You are watching fireworks over an iego ay as you sail away in a boat. he highest point the fireworks reach is about 0. ile above the bay. Your eyes are about 0.0 ile above the water. t point you can no longer see the fireworks because of the curvature of arth. he radius of arth is about 4000 iles, and is tangent to arth at point. ind. Round your answer to the nearest tenth. 9. ROO In the diagra, and N are secant lines that intersect at point. rove that N > N. N Not drawn to scale 5. HI ONNION In the diagra, is tangent to. Write an algebraic expression for in ters of x. hen find. 7 3 40 6. HI ONNION he circles in the diagra are concentric. Write an algebraic expression for c in ters of a and b. a b c 30. IN N RUN Your friend clais that it is possible for a circuscribed angle to have the sae easure as its intercepted arc. Is your friend correct? xplain your reasoning. 3. RONIN oints and are on a circle, and t is a tangent line containing and another point. a. raw two diagras that illustrate this situation. b. Write an equation for in ters of for each diagra. c. or what easure of can you use either equation to find? xplain. 3. RONIN YZ is an equilateral triangle inscribed in. is tangent to at point, is tangent to at point Y, and is tangent to at point Z. raw a diagra that illustrates this situation. hen classify by its angles and sides. ustify your answer. ection 0.5 ngle Relationships in ircles 57
33. ROVIN HOR o prove the angent and Intersected hord heore (heore 0.4), you ust prove three cases. a. he diagra shows the case where contains the center of the circle. Use the angent ine to ircle heore (heore 0.) to write a paragraph proof for this case. 36. HOUH ROVOIN In the figure, and are tangent to the circle. oint is any point on the ajor arc fored by the endpoints of the chord. abel all congruent angles in the figure. ustify your reasoning. b. raw a diagra and write a proof for the case where the center of the circle is in the interior of. c. raw a diagra and write a proof for the case where the center of the circle is in the exterior of. 34. HOW O YOU I? In the diagra, television caeras are positioned at and to record what happens on stage. he stage is an arc of. You would like the caera at to have a 30 view of the stage. hould you ove the caera closer or farther away? xplain your reasoning. 37. ROVIN HOR Use the diagra below to prove the ngles Outside the ircle heore (heore 0.6) for the case of a tangent and a secant. hen copy the diagras for the other two cases on page 567 and draw appropriate auxiliary segents. Use your diagras to prove each case. 38. ROVIN HOR rove that the ircuscribed ngle heore (heore 0.7) follows fro the ngles Outside the ircle heore (heore 0.6). 80 30 5 In xercises 39 and 40, find the indicated easure(s). ustify your answer. 39. ind when WZY = 00. W 40. ind and. Z Y 60 0 35. ROVIN HOR Write a proof of the ngles Inside the ircle heore (heore 0.5). iven hords and intersect inside a circle. rove = ( + ) 5 H 85 aintaining atheatical roficiency olve the equation. (kills Review Handbook) Reviewing what you learned in previous grades and lessons 4. x + x = 4. x = x + 35 43. 3 = x + 4x 57 hapter 0 ircles