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10.4 Use Inscribed Angles and Polygons Goal Your Notes p Use inscribed angles of circles. VOCABULARY Inscribed angle Intercepted arc Inscribed polygon Circumscribed circle THEOREM 10.7: MEASURE OF AN INSCRIBED ANGLE THEOREM A D C The measure of an inscribed angle is one half the measure of its 1 m∠ADB 5 } 2 intercepted arc. B Use inscribed angles Example 1 Find the indicated measure in (P. C a. m∠S R b. mRQ 608 Q 378 P T S Solution 1 a. m∠S 5 } 2 1 5} ( 2 )5 C 52p 5 Because C RQS is a semicircle, RQ 5 1808 2 5 1808 2 mC b. mQS 5 2m∠ 278 Lesson 10.4 • Geometry Notetaking Guide . 5 . Copyright © Holt McDougal. All rights reserved. 10.4 Use Inscribed Angles and Polygons Goal Your Notes p Use inscribed angles of circles. VOCABULARY Inscribed angle An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. Intercepted arc The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc of the angle. Inscribed polygon A polygon is an inscribed polygon if all of its vertices lie on a circle. Circumscribed circle A circumscribed circle is a circle that contains the vertices of an inscribed polygon. THEOREM 10.7: MEASURE OF AN INSCRIBED ANGLE THEOREM A D C The measure of an inscribed angle B is one half the measure of its 1 m∠ADB 5 } mAB 2 intercepted arc. C Use inscribed angles Example 1 Find the indicated measure in (P. C a. m∠S R b. mRQ Solution 1 C 608 Q 378 P T S 1 a. m∠S 5 } mRT 5 } ( 608 ) 5 308 2 2 C Because C RQS is a semicircle, RQ 5 1808 2 mC QS 5 1808 2 mC b. mQS 5 2m∠ R 5 2 p 378 5 748 . 278 Lesson 10.4 • Geometry Notetaking Guide 748 5 1068 . Copyright © Holt McDougal. All rights reserved. Your Notes Example 2 C Find the measure of an intercepted arc Find mHJ and m∠HGJ. What do you notice about ∠HGJ and ∠HFJ ? G C 1 1 5} ( 2 J . )5 THEOREM 10.8 . A D If two inscribed angles of a circle intercept the same arc, then the angles are congruent. Example 3 398 F Solution From Theorem 10.7, you know that 5 2( )5 mHJ 5 2m∠ Also, m∠HGJ 5 } 2 ∠HFJ. So ∠HGJ H B C ∠ADB > ∠ Use Theorem 10.8 Q Name two pairs of congruent angles in the figure. P Solution S Notice that ∠QRP and ∠ intercept the same arc, and so ∠QRP > ∠ by Theorem 10.8. Also, ∠RQS and ∠ intercept the same arc, so ∠RQS > ∠ . R Checkpoint Find the indicated measure. C 1. m∠GHJ 2. mCD H B 1008 Copyright © Holt McDougal. All rights reserved. R C 408 F G 3. m∠RTS Q J D 318 S T Lesson 10.4 • Geometry Notetaking Guide 279 Your Notes Example 2 C Find the measure of an intercepted arc Find mHJ and m∠HGJ. What do you notice about ∠HGJ and ∠HFJ ? G 1 C 398 F Solution From Theorem 10.7, you know that mHJ 5 2m∠ HFJ 5 2( 398 ) 5 788 . C H J 1 Also, m∠HGJ 5 } mHJ 5 } ( 788 ) 5 398 . 2 2 So ∠HGJ > ∠HFJ. THEOREM 10.8 A D If two inscribed angles of a circle intercept the same arc, then the angles are congruent. Example 3 B C ∠ADB > ∠ ACB Use Theorem 10.8 Q Name two pairs of congruent angles in the figure. P Solution S Notice that ∠QRP and ∠ QSP intercept the same arc, and so ∠QRP > ∠ QSP by Theorem 10.8. Also, ∠RQS and ∠ RPS intercept the same arc, so ∠RQS > ∠ RPS . R Checkpoint Find the indicated measure. C 1. m∠GHJ 2. mCD H B 1008 Q J m∠GHJ 5 508 Copyright © Holt McDougal. All rights reserved. R C 408 F G 3. m∠RTS C D mCD 5 808 318 S T m∠RTS 5 318 Lesson 10.4 • Geometry Notetaking Guide 279 Your Notes THEOREM 10.9 If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle. Example 4 A D B C m∠ABC 5 908 if and only if is a diameter of the circle. Use a circumscribed circle Security A security camera rotates 908 and needs to be able to view the width of a wall. The camera is placed in a spot where the only thing viewed when rotating is the wall. You want to change the camera’s position. Where else can it be placed so that the wall is viewed in the same way? Solution From Theorem 10.9, you know that if a right triangle is inscribed in a circle, then the hypotenuse of the triangle is of the circle. So, draw the a circle that has the width of the wall as . The wall fits perfectly with a your camera’s 908 rotation from any in front of point on the the wall. THEOREM 10.10 E A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. D, E, F, and G lie on (C if and only if m∠D 1 m∠F 5 m∠E 1 m∠G 5 280 Lesson 10.4 • Geometry Notetaking Guide F C D G . Copyright © Holt McDougal. All rights reserved. Your Notes THEOREM 10.9 If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle. Example 4 A D B C m∠ABC 5 908 if } and only if AC is a diameter of the circle. Use a circumscribed circle Security A security camera rotates 908 and needs to be able to view the width of a wall. The camera is placed in a spot where the only thing viewed when rotating is the wall. You want to change the camera’s position. Where else can it be placed so that the wall is viewed in the same way? Solution From Theorem 10.9, you know that if a right triangle is inscribed in a circle, then the hypotenuse of the triangle is a diameter of the circle. So, draw the circle that has the width of the wall as a diameter . The wall fits perfectly with your camera’s 908 rotation from any point on the semicircle in front of the wall. THEOREM 10.10 A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. D, E, F, and G lie on (C if and only if m∠D 1 m∠F 5 m∠E 1 m∠G 5 1808 . 280 Lesson 10.4 • Geometry Notetaking Guide E F C D G Copyright © Holt McDougal. All rights reserved. Your Notes Example 5 Use Theorem 10.10 Find the value of each variable. a. b. Q 3y8 888 P 1008 K J 8x 8 y8 R 4x8 L 3y8 x8 M S Solution a. PQRS is inscribed in a circle, so opposite angles . are m∠P 1 m∠R 5 m∠Q 1 m∠S 5 1008 1 y8 5 888 1 x8 5 y5 x5 b. JKLM is inscribed in a circle, so opposite angles are . m∠ J 1 m∠L 5 m∠K 1 m∠M 5 8x8 1 4x8 5 3y8 1 3y8 5 12x 5 6y 5 x5 y5 Checkpoint Complete the following exercises. 4. A right triangle is inscribed in a circle. The radius of the circle is 5.6 centimeters. What is the length of the hypotenuse of the right triangle? Homework 5. Find the values of a and b. B A 5a8 4b8 C 968 Copyright © Holt McDougal. All rights reserved. D Lesson 10.4 • Geometry Notetaking Guide 281 Your Notes Example 5 Use Theorem 10.10 Find the value of each variable. a. b. Q 3y8 888 P 1008 K J 8x 8 y8 R 4x8 L 3y8 x8 M S Solution a. PQRS is inscribed in a circle, so opposite angles are supplementary . m∠P 1 m∠R 5 1808 m∠Q 1 m∠S 5 1808 1008 1 y8 5 1808 888 1 x8 5 1808 y 5 80 x 5 92 b. JKLM is inscribed in a circle, so opposite angles are supplementary . m∠ J 1 m∠L 5 1808 m∠K 1 m∠M 5 1808 8x8 1 4x8 5 1808 3y8 1 3y8 5 1808 12x 5 180 6y 5 180 x 5 15 y 5 30 Checkpoint Complete the following exercises. 4. A right triangle is inscribed in a circle. The radius of the circle is 5.6 centimeters. What is the length of the hypotenuse of the right triangle? 11.2 centimeters Homework 5. Find the values of a and b. a 5 18, b 5 21 B A 5a8 4b8 C 968 Copyright © Holt McDougal. All rights reserved. D Lesson 10.4 • Geometry Notetaking Guide 281