New Vocabulary inscribed angle. At the right, the vertex of &C is on O, and the sides of &C are chords of the circle. &C is an

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1 -. lan - Inscribed ngles bjectives o find the measure of an inscribed angle o find the measure of an angle formed by a tangent and a chord amples Using the Inscribed ngle heorem Using orollaries to Find ngle Measures Using heorem - What You ll Learn o find the measure of an inscribed angle o find the measure of an angle formed by a tangent and a chord... nd Why o use theorems and corollaries to find missing lengths in circles, as in ample Identify the following in at the right.. a semicircle. a minor arc. a major arc 4. a central angle l Find the measure of each arc in New Vocabulary inscribed angle 4. nswers may vary. amples are given. heck kills You ll Need G for Help intercepted arc Lesson -6 Math ackground orollary to the Inscribed ngle heorem suggests this method of constructing a right angle: onstruct a circle of any radius with any center, then use a straightedge to draw a diameter. ny angle with verte on the circle whose sides intersect the endpoints of the diameter is a right angle. More Math ackground: p. 66 Finding the Measure of an Inscribed ngle Hands-n ctivity: ploring Inscribed ngles raw two large circles with a compass. Label the centers X and Y. n the circles, use a straightedge and copy the diagrams shown. X 4 Y 5 6 Lesson lanning and esources ee p. 66 for a list of the resources that support this lesson. oweroint ell inger ractice heck kills You ll Need For intervention, direct students to: entral ngles and rcs Lesson -6: amples, tra kills, Word roblems, roof ractice, h.. a. heck students work. b. ml m c. ml ml ml. a. heck students work. b. he measure of an l whose verte is on a circle and whose sides intersect the endpts. of a diameter of the circle is 9.. a. atterns In X, use a protractor to measure &X and each numbered angle. etermine m. ecord your results and look for patterns. ompare your results with others. a-c. ee left. b. Write a conjecture about the relationship between m& and m. c. Write a conjecture about the measures of &, &, and &.. a. atterns Use a protractor to measure the numbered angles in Y. ecord your results and look for patterns. ompare your results. b. Write a conjecture about an angle whose verte is on a circle and whose sides intersect the endpoints of a diameter of the circle. a-b. ee left. t the right, the verte of & is on, and the sides of & are chords of the circle. & is an inscribed angle. is the intercepted arc of &. Intercepted arc Inscribed angle 678 hapter ircles 678 pecial Needs L Have students use colored pencils to trace intercepted arcs from one endpoint to the other. his helps students identify arcs intercepted by inscribed angles, especially when several chords are shown. learning style: visual elow Level L he ctivity can be done using geometry software. For part, fi points and and move point on. he measures of and as point moves illustrate heorem -9. learning style: tactile

2 heorem -9 describes the relationship between an inscribed angle and its intercepted arc.. each Key oncepts heorem -9 Inscribed ngle heorem Vocabulary ip When different conditions are possible, each possibility can be called a case. You can prove a theorem by proving it for all possible cases. roof uick heck o prove heorem -9, there are three cases to consider. proof of ase I is below. You will prove ases II and III in ercises 6 and 7. roof of heorem -9, ase I Given: with inscribed & and diameter rove: m& = m raw radius to form isosceles # with = and, hence, m& = m&. m = m& efinition of m m = m& + m& riangle terior ngle heorem = m& ubstitute and simplify. = m& olve for ml. You can use the Inscribed ngle heorem to find missing measures in circles. XML I: he center is on a side of the angle. Using the Inscribed ngle heorem Find the values of a and b. m& = m Inscribed ngle heorem 6 = a ubstitute. = a olve for a. m& = m Inscribed ngle heorem b = m m rc ddition ostulate = ( ) ubstitute. b = 75 implify. Find m& if m = 6. dvanced Learners L4 Have students use a right triangle inscribed in a circle to prove that the median drawn to the hypotenuse of a right triangle forms two isoceles triangles. he measure of an inscribed angle is half the measure of its intercepted arc. m& = m II: he center is inside the angle. 5 III: he center is outside the angle. Lesson - Inscribed ngles nglish Language Learners LL Make sure students do not confuse the measure of an inscribed angle is half the measure of its intercepted arc with the measure of an intercepted arc is half the measure of its inscribed angle. Guided Instruction Hands-n ctivity tudents will measure and make conjectures about the relationships between inscribed angles and their intercepted arcs. oint out that the placement of the vertices in Y is unimportant when copying the diagrams but that the sides of angles 4, 5, and 6 must intersect the endpoints of the diameter. Math ip he Inscribed ngle heorem must be proved for three mutually eclusive cases: the center of the circle on a side of the angle, in the interior of the angle, and eterior to, or outside the angle. sk students to restate the rc ddition ostulate. emind them of its similarity to the ngle ddition ostulate. echnology ip Have students use geometry software to eplore the corollaries. rror revention! tudents may carelessly apply the Inscribed ngle heorem to angles whose vertices are not on the circle. emind them that the theorem applies only to angles whose vertices are on a circle and whose sides contain chords of the circle. XML XML oweroint dditional amples Find the values of and y. 8 y 9 G 75; y 95 7 F learning style: verbal learning style: verbal 679

3 oweroint dditional amples You will use three corollaries to the Inscribed ngle heorem to find measures of angles in circles. You will justify these corollaries in ercises 8, 9, and 4. Find the values of a and b. b Key oncepts orollaries orollaries to the Inscribed ngle heorem. wo inscribed angles that intercept the same arc are congruent.. n angle inscribed in a semicircle is a right angle.. he opposite angles of a quadrilateral inscribed in a circle are supplementary. a 9; b 6 Guided Instruction XML lternative Method tudents can find the value of by applying orollary to the Inscribed ngle heorem and then using the riangle ngle-um heorem to find the value of y. oweroint dditional amples and are diameters of * ) U. is tangent to at point. Find m& and m&. ml 7; ml = 6 esources aily Notetaking Guide - L aily Notetaking Guide - dapted Instruction L losure a 6 U For: Inscribed ngle ctivity Use: Interactive etbook, - uick heck XML Using orollaries to Find ngle Measures Find the measure of the numbered angle. a. b. 4 & is inscribed in a semicircle. & and the 88 angle intercept the y orollary, & is a right angle. same arc. y orollary, the angles m& = 9 are congruent, so m& = 8. For the diagram at the right, find the measure of each numbered angle. ml 9, ml, ml 9, ml4 7. In the diagram, and are fied points, and point moves along the circle. From the Inscribed ngle heorem, you know that as moves, m& remains the same and is m. s the last diagram suggests, this is also true when and coincide. 7 he ngle Formed by a angent and a hord Key oncepts heorem - he measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. m& = m In the diagram, is tangent to at point, and is tangent to at point. Find m. 68 hapter ircles You will prove heorem - in ercise

4 uick heck XI ample (page 679) XML Using heorem ) - In the diagram at the right, KJ is tangent to the circle at J. Find the values of and y. = mjl heorem - = m& Inscribed ngle heorem = 5 ubstitution y = mj heorem - = m - mjl L rc ddition ostulate = (8-7) ubstitute. = 55 implify. escribe two ways to find m&jk using heorem -. mljk mlljk ± mljl 5 ± 9 5 mljk mlj (7 ± 8) 5 For more eercises, see tra kill, Word roblem, and roof ractice. ractice and roblem olving G ractice by ample for Help 4a. l and ; l and ; l and ; l and Identify the inscribed angle and its intercepted arc. lmn; MN... M L N l; l; 4. a. Name the four inscribed angles and their intercepted arcs. b. Which angles appear to intercept major arcs? What kind of angles do these appear to be? l and l; obtuse 5 y L J K. ractice ssignment Guide -4, -, , hallenge 4-45 est rep 46-5 Mied eview 5-58 Homework uick heck o check students understanding of key skills and concepts, go over ercises, 6,, 5, 5. ercise 4 eview the definitions of semicircle, minor arc, and major arc and how to name these arcs. ercise 8 tudents who focus on theorems from this lesson to answer the eercises may overlook the Vertical ngles heorem. Use this eercise as an opportunity to remind them that it is often necessary to integrate earlier work. 8. a 54; b ; c a ; b ; c 8. a ; b 67; c 84; d 8. 6; y 6. a 85; b 47.5; c 9. a 5; b 9; c 9 4. p 9; q Find the value of each variable ee left a 8; b d G Guided roblem olving nrichment eteaching dapted ractice ractice Name lass ate ractice - egular octagon IGHU is divided into eight congruent triangles. Find the image of each point or segment for the given rotation.. 45 rotation of G about Z. 5 rotation of U about Z. 5 rotation of about Z 4. 7 rotation of I about Z U L L4 L L L otations I G Z H 5. 5 rotation of about Z ample (page 68) p q y Lesson - Inscribed ngles 68 earson ducation, Inc. ll rights reserved rotation of about Z opy each figure and point. raw the image of each figure for the given rotation about. Label the vertices of each image N U. otate the heagon 6 about point, draw its image, and label the vertices. epeat this procedure three more times, rotating the original figure, 8, and 4. M N L. In ercise, which verte is closest to point in all four figures? opy k and point. hen draw the image for the given composition of rotations about point.. and then 7. and then 68

5 ercises, tudents who have trouble identifying the arcs intercepted by inscribed angles when several chords are shown may copy the diagram for each part of the eercise, coloring only the inscribed angle and the intercepted arc whose measure is asked for. ercise ncourage students to write an algebraic statement for m such as (5 ) = m + 7. ercise 5 fter students complete the eercise, ask: How can you state the result as a theorem? ample: In a circle, arcs between parallel chords are congruent. ercise 6 fter students complete this eercise, have them use their eplanations in ercise 5 to justify their conjectures. ercise 8 efore assigning this eercise, ask students to sketch several different inscribed parallelograms as they can in a circle. ample (page 68) 8.a 6; b 64; c 4 9.a ; b 78; c 56.a ; b 6; c 6; d 4; e 6 pply Your kills Find the value of each variable. You may assume that rays that appear to be tangent are tangent y 7. f e 5 w 65; y e 65; f 8. m 9.. d 5 6 e Find each indicated measure for.. a. m a. m& 4 b. m& 55 b. m c. m& c. m& 77 4 d. m 54 d. m& 4 e. m& Multiple hoice What is the measure of in the 7 circle at the right? Find each indicated measure for the circle in ercise. 5 a. m 78 b. m& 95 c. m& 5 d. m& l l since they are alt. int. '. ince inscribed ' intercept arcs, m m. 6a. heck students work. 5. Writing opy the diagram at the right on your paper. raw chord. plain why m = m. 6. a. pen-nded ketch a trapezoid inscribed in a circle. epeat several times using different circles. b. Make a onjecture What kind of trapezoid can be inscribed in a circle? Justify your response. ee margin. Vocabulary ip regular star (ercise 7) has congruent sides. lso, the angles pointing out are congruent, as are the angles pointing in. 7. Landscape rchitecture ome circular nglish gardens, like the one shown here, have paths in the shape of an inscribed regular star. a. Find the measure of an inscribed angle formed by the star in the garden shown here b. What is the measure of an inscribed angle in a garden with a five-pointed star? 6 9. about 7. cm by 7. cm. about 4. cm for each side. about 7. cm legs, and a cm base 8. ritical hinking parallelogram inscribed in a circle must be what kind of parallelogram? plain. ee margin. Graphing alculator he diameter of a circle is cm. Find the dimensions of the largest figure of each type that can be inscribed in the circle. (Hint: Use techniques demonstrated in the ploration on page 66.) 9. a rectangle. a triangle. a right triangle 68 hapter ircles 6.b. isosc. trapezoid; justifications may vary. ample: rcs between two n chords are ectangle; opposite ' are (because figure is $) and suppl. (because opp. ' intercept arcs whose measures sum to 6). ongruent suppl. ' are rt ', so inscribed $ must be a rectangle.. nswers may vary. ample: a. If the cameras lenses open at ', then in the positions shown they share the same arc of the scene. b. No; the distances from each position of the scene to each camera affect the look of the scene.

6 . elevision he director of a telecast wants the option of showing the same scene from three different views. a-b. ee margin p. 68. a. plain why cameras in the positions shown in the diagram will transmit the same scene. b. ritical hinking Will the scenes look the same to the director when she views them on the control room monitors? plain. amera cene amera amera ercises 6 4 Given the importance of the results, you may want to prove these as a class or ask volunteers to prepare the proofs before class and present them on the board for the class to discuss and critique. ach proof should carefully state whether a chord passes through the center. eal-world onnection big red tally light on each camera tells this anchorman which camera is hot (on the air).. onstructions he diagrams below show the construction of a tangent to a circle from a point outside the circle. plain why must be tangent to. (Hint: opy the third diagram and draw. ) ee margin. * ) Given: and point. onstruct the midpoint of. Label the point. onstruct a semicircle with radius and center. Label its intersection with as. raw. G F G nline Homework Help Visit: Hchool.com Web ode: aue- 8.. (, l intercepts, and l intercepts (Given). ml m and ml m (Inscr. l hm.). ml ml (ubst.) 4. l l (ef. of ). l is a rt. l because it is inscribed in semicircle, and if a line is # to a radius at its endpoint, it is tangent to the circle. 4. echnology onstruct and the chords shown with geometry software. a. s you move on between and, which inscribed angles remain congruent? lf, lfg, and lg b. Which inscribed angle remains a right angle? l c. Which inscribed angles remain supplementary in quadrilateral FG? lfg and lg; lf and lfg 5. onstructions Use orollary of heorem -9 to construct a right triangle G given one leg and the hypotenuse. ee margin. roof Write a proof for each of ercises Inscribed ngle heorem, ase II 7. Inscribed ngle heorem, ase III Given: with inscribed & Given: with inscribed & rove: m& = m rove: m& = m Hint: Use the Hint: Use the Inscribed ngle Inscribed ngle heorem, ase I. heorem, ase I ee margin. 8. Inscribed ngle heorem, or. 9. Inscribed ngle heorem, or. Given: ; & intercepts, Given: with & inscribed and & intercepts. in a semicircle ee margin. rove: & > & ee left. rove: & is a right angle. Lesson - Inscribed ngles ( with inscribed l (Given) leg. ml m ; hypotenuse ml m (Inscribed l hm., ase I). ml ± ml ml (l dd. ost.) 4. m ± m ml (ubst.) 5. (m ± m ) ml (istr. rop.) 6. m ml (rc dd. ost.) 7.. ( with inscribed l (Given). ml m (Inscribed l hm., ase I). ml m (Inscr. l hm., ase I) 4. m m m (rc dd. ost.) 5. ml ml ml (l dd. ost.) 6. ml m m (ubst.) 7. ml m (ubst.) 9.. ( with inscribed l in a semicircle (Given). ml m (Inscr. l hm.). m 8 (Meas. of semicircle 8.) 4. ml 9 (ubst.) 5. l is a rt. l. (ef. of rt. l) 68

7 4. ssess & eteach oweroint Lesson uiz In the diagram below, circumscribes quadrilateral and is inscribed in quadrilateral XYZW. Y X 6 9 W Z. Find the measure of each inscribed angle. ml ; ml 75; ml 8; ml 5. Find m&z. 45. re &X and &X congruent? plain. Yes; each is formed by a tangent and a chord, and they intercept the same arc. 4. Find the angle measures in quadrilateral XYZW. mlx 8; mly 7; mlz 9; mlw 5. oes a diagonal of quadrilateral intersect the center of the circle? plain how you can tell. No; the diagonal would be a diameter of and the inscribed angle would be a right angle, which was not found in ercise above. lternative ssessment Give each student a compass and straightedge. Have students use the theorems in Lesson - to plan, justify, and eecute a method for constructing a right angle. roblem olving Hint In ercise 4, let GH first be a diameter. hallenge est rep Multiple hoice hort esponse tended esponse 4. Inscribed ngle heorem, or. 4. heorem - Given: quadrilateral Given: GH and tangent / inscribed in intersecting at H on rove: & and & are supplementary. rove: m&ghi = mgfh & and & are supplementary. ee margin. ee back of book. F G I H ritical hinking Is the statement true or false? If true, give a convincing argument. If false, give a countereample. ee margin. 4. If two angles inscribed in a circle are congruent, then they intercept the same arc. 4. If an inscribed angle is a right angle, then it is inscribed in a semicircle. true 44. circle can always be circumscribed about a quadrilateral whose opposite angles are supplementary. true 45. onstructions raw two segments. Label their lengths and y. onstruct the geometric mean of and y.(hint: ecall a theorem about a geometric mean.) ee margin. In ercises 46 and 47, what is the value of each variable? F.. 5 J y G.. 45 H J In the figure at the right, a square is circumscribed about. What is the area of the square?. 64 in.. 9 in.. 56 in.. (56 + 6Á) in. 49. a. plain how you can find m&xyz. b. Find m&xyz. a-b. ee margin p Use the figure at the right. a-c. ee margin p a. What is m&? plain. b. What is m&? plain. c. Use an equation to find the value of. W 4 8 in. 54 X Y 7 Z 684 hapter ircles 4.. GH and tangent < intersecting at H on ( (Given). onstruct diameter H intersecting circle at. (onstr.). lhi is a rt. l. 684 (angent and radius are #.) 4. GH is a semicircle of measure 8. (ef. of semicircle) 5. mlhg ± mlghi mlhi (l dd. ost.) 6. m G ± mgfh m GH (rc dd. ost.) 7. 9 mlhg ± mlghi (ubst.) 8. 8 m G ± m GFH (ubst.) 9. 9 (m ± G m GFH) (iv. rop.). mlhg ± mlghi m G ± mgfh (ubst. and istr. rop.). mlhg m G (Inscr. l hm.). mlghi mgfh (ubtr. rop.)

8 Mied eview est rep G for Help Lesson - lgebra Find the value of to the nearest tenth esources For additional practice with a variety of test item formats: tandardized est rep, p. 7 est-aking trategies, p. 76 est-aking trategies with ransparencies Lesson -5 Lesson 7-57.oth have rt. ' and the vertical ' are, so > are M by M ost Find the area of each triangle. Give answers to the nearest tenth in. cm 8 m in. 5 cm 49. in cm Indirect Measurement o find the width of a river, you have made the measurements shown in the sketch. 57. plain why the triangles are similar. 58. a. Find the width of the river in feet. 96 ft b. Find the width of the river in miles. about.8 mi d 4 ft 49.9 m 4 m 75 ft ft Use this heckpoint uiz to check students understanding of the skills and concepts of Lessons - through -. esources Grab & Go heckpoint uiz heckpoint uiz Lessons - through - 7. w 4; ; y 8 8. a ; b 4; c 8; d 6 9. w 5; 75; y.a 4; b 7; c 47.5 ach polygon below circumscribes the circle. Find the perimeter of the polygon.. 76 cm 9 cm. in.. 5 m m in. 7.5 m 4 in. cm lgebra Find the value of lesson quiz, Hchool.com, Web ode: aua- 9 6 cm 5 m 8 in. 7 m 48 in. 5 m lgebra Find the value of each variable. Lines that appear to be tangent are tangent. 7-. ee left y. 5 w 44 y w 6 54 d false 45. In the construction below,, y, is the mdpt. of, and #. 7 Lesson - Inscribed ngles [] a. m WY 4; m WX? 54 8; mxy m&xyz. hus, m&xyz 56. b. 56 [] incorrect answer incorrect eplanation 5. [4] a. m 4, so m, and l. b. ml 9, because it is inscribed in a semicircle. c. 8 ± 4. hus, 7 4 and 7.5. [] appropriate methods, but with one computational error [] incorrect methods solved correctly correct methods solved incorrectly [] correct answers to a c, without work shown 685