Objectives To find the measure of an inscribed angle To find the measure of an angle formed by a tangent and a chord

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1 1-3 Inscribed ngles ommon ore State Standards G-.. Identify and describe relationships among inscribed angles, radii, and chords. lso G-..3, G-..4 M 1, M 3, M 4, M 6 bjectives To find the measure of an inscribed angle To find the measure of an angle formed by a tangent and a chord raw a large diagram and draw the angle each point makes with the goal posts. Three high-school soccer players practice kicking goals from the points shown in the diagram. ll three points are along an arc of a circle. layer says she is in the best position because the angle of her kicks toward the goal is wider than the angle of the other players kicks. o you agree? Explain. layer MTHEMTIL RTIES layer layer Lesson Vocabulary inscribed angle intercepted arc n angle whose vertex is on the circle and whose sides are chords of the circle is an inscribed angle. n arc with endpoints on the sides of an inscribed angle, and its other points in the interior of the angle is an intercepted arc. In the diagram, inscribed intercepts. Intercepted arc Inscribed angle Essential Understanding ngles formed by intersecting lines have a special relationship to the arcs the intersecting lines intercept. In this lesson, you will study arcs formed by inscribed angles. Theorem 1-11 Inscribed ngle Theorem The measure of an inscribed angle is half the measure of its intercepted arc. m = 1 m 780 hapter 1 ircles

2 To prove Theorem 1-11, there are three cases to consider. I: The center is on a side of the angle. II: The center is inside the angle. III: The center is outside the angle. elow is a proof of ase I. You will prove ase II and ase III in Exercises 6 and 7. of Theorem 1-11, ase I Given: } with inscribed and diameter rove: m = 1 m raw radius to form isosceles with = and, hence, m = m (Isosceles Triangle Theorem). m = m + m Triangle Exterior ngle Theorem m = m efinition of measure of an arc m = m + m Substitute. m = m Substitute and simplify. 1 m = m ivide each side by. Which variable should you solve for first? You know the inscribed angle that intercepts T, which has the measure a. You need a to find b. So find a first. roblem 1 What are the values of a and b? m QT = 1 m T Using the Inscribed ngle Theorem 60 = 1 a Substitute. Inscribed ngle Theorem 10 = a Multiply each side by. m RS = 1 m S Inscribed ngle Theorem m RS = 1 (m T + m TS ) rc ddition ostulate Q 60 R T 30 S b = 1 ( ) Substitute. b = 75 Simplify. Got It? 1. a. In }, what is m? b. What are m, m, m, and m? c. What do you notice about the sums of the measures of the opposite angles in the quadrilateral in part (b)? Lesson 1-3 Inscribed ngles 781

3 You will use three corollaries to the Inscribed ngle Theorem to find measures of angles in circles. The first corollary may confirm an observation you made in the Solve It. orollaries to Theorem 1-11: The Inscribed ngle Theorem orollary 1 Two inscribed angles that intercept the same arc are congruent. orollary n angle inscribed in a semicircle is a right angle. orollary 3 The opposite angles of a quadrilateral inscribed in a circle are supplementary. You will prove these corollaries in Exercises Is there too much information? Each diagram has more information than you need. Focus on what you need to find. roblem Using orollaries to Find ngle Measures What is the measure of each numbered angle? is inscribed in a semicircle. and the 38 angle intercept the y orollary, 1 is a right angle, so same arc. y orollary 1, the angles m 1 = 90. are congruent, so m = 38. Got It?. In the diagram at the right, what is the measure of each numbered angle? The following diagram shows point moving along the circle until a tangent is formed. From the Inscribed ngle Theorem, you know that in the first three diagrams m is 1 m. s the last diagram suggests, this is also true when and coincide. 78 hapter 1 ircles

4 Theorem 1-1 The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. m = 1 m You will prove Theorem 1-1 in Exercise 34. roblem 3 Using rc Measure In the diagram, < SR > is a tangent to the circle at Q. If mmq = 1, what is mjqr? S Q < SR > is tangent to the circle at Q m MQ = 1 M R m QR m QS + m QR = 180. So first find m QS using MQ. How can you check the answer? ne way is to use m QR to find m Q. onfirm that m Q + m MQ = mmq = m QS 1 (1) = m QS Substitute. 106 = m QS Simplify. m QS + m QR = 180 Linear air ostulate m QR = 180 Substitute. m QR = 74 Simplify. The measure of an formed by a tangent and a chord is 1 the measure of the intercepted arc. Got It? 3. a. In the diagram at the right, KJ is tangent to }. What are the values of x and y? b. Reasoning In part (a), an inscribed angle ( Q) and an angle formed by a tangent and chord ( KJL) intercept the same arc. What is always true of these angles? Explain. Q 35 y L J x K Lesson 1-3 Inscribed ngles 783

5 Lesson heck o you know HW? Use the diagram for Exercises Which arc does intercept?. Which angle intercepts? 3. Which angles of quadrilateral are supplementary? MTHEMTIL RTIES o you UNERSTN? 4. Vocabulary What is the relationship between an inscribed angle and its intercepted arc? 5. Error nalysis classmate says that m = 90. What is your classmate s error? ractice and roblem-solving Exercises MTHEMTIL RTIES ractice Find the value of each variable. For each circle, the dot represents the center. See roblems 1 and d x 7 y p 58 q Find the value of each variable. Lines that appear to be tangent are tangent. See roblem y x w e f pply 19. Writing parallelogram inscribed in a circle must be what kind of parallelogram? Explain. 784 hapter 1 ircles

6 Find each indicated measure for. 0. a. m a. m b. m b. m E 48 c. m c. m d. m d. m e. m E E. Think bout a lan What kind of trapezoid can be inscribed in a circle? Justify your response. raw several diagrams to make a conjecture. How can parallel lines help? Find the value of each variable. For each circle, the dot represents the center e d 56 Write a proof for Exercises 6 and Inscribed ngle Theorem, ase II Given: } with inscribed rove: m = 1 m (Hint: Use the Inscribed ngle Theorem, ase I.) 7. Inscribed ngle Theorem, ase III Given: }S with inscribed QR rove: m QR = 1 m R R S T (Hint: Use the Inscribed ngle Theorem, ase I.) 8. Television The director of a telecast wants the option of showing the same scene from three different views. a. Explain why cameras in the positions shown in the diagram will transmit the same scene. b. Reasoning Will the scenes look the same when the director views them on the control room monitors? Explain. Q Scene amera 1 amera 3 amera Lesson 1-3 Inscribed ngles 785

7 9. Reasoning an a rhombus that is not a square be inscribed in a circle? Justify your answer. 30. onstructions The diagrams below show the construction of a tangent to a circle from a point outside the circle. Explain why < > must be tangent to }. (Hint: opy the third diagram and draw.) Given: and point onstruct the midpoint of. Label the point. onstruct a semicircle with radius and center. Label its intersection with as. raw. Write a proof for Exercises Inscribed ngle Theorem, orollary 1 Given: }, intercepts, intercepts. rove: 33. Inscribed ngle Theorem, orollary 3 Given: Quadrilateral inscribed in } rove: and are supplementary. and are supplementary. 3. Inscribed ngle Theorem, orollary Given: } with inscribed in a semicircle rove: is a right angle. 34. Theorem 1-1 Given: GH and tangent / intersecting }E at H rove: m GHI = 1 m GFH G E F H I hallenge Reasoning Is the statement true or false? If it is true, give a convincing argument. If it is false, give a counterexample. 35. If two angles inscribed in a circle are congruent, then they intercept the same arc. 36. If an inscribed angle is a right angle, then it is inscribed in a semicircle. 37. circle can always be circumscribed about a quadrilateral whose opposite angles are supplementary. 786 hapter 1 ircles

8 38. rove that if two arcs of a circle are included between parallel chords, then the arcs are congruent. 39. onstructions raw two segments. Label their lengths x and y. onstruct the geometric mean of x and y. (Hint: Recall a theorem about a geometric mean.) ERFRMNE TSK pply What You ve Learned Look back at the information given on page 761 about the logo for the showroom display. The diagram of the logo is shown again below. onsider relationships of angles and arcs in the diagram. Select all of the following that are true. Explain your reasoning. MTHEMTIL RTIES M 1 9 ft E G 7. ft 15 ft 7 ft. is an inscribed angle in }.. G is an inscribed angle in }.. G intercepts G.. G intercepts E. E. The measure of G is half the measure of G. F. G is a right triangle. G. G G Lesson 1-3 Inscribed ngles 787