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Name Date Class Practice B LESSON 11-4 Inscribed Angles Find each measure. " 1. # ! mDEA 1 ' 2. 130° 138° mQRS 3 0 mTSR ( 6 4. 10° 90.5° mXVU mVXW 4 mGH ) 2 9° 78° mFGI & % $ 3. 33° 192° mCED 5 8 . 7 , 5. A circular radar screen in an air traffic control tower shows these flight paths. Find mLNK. 73° + . - Find each value. * " 6. & 7. X Y % X ) $ # 48° mCED , 8. - 13 y 9. 6 a A ' ( 4 1 * B mSRT 77° 2 + nB 3 Find the angle measures of each inscribed quadrilateral. mX 9 X 10. 8 X : X 5 4 mZ mW 7 12. mY mT Z 6 mU Z mV Z 7 mW Copyright © by Holt, Rinehart and Winston. All rights reserved. 71° 109° 109° 71° 68° 95° 112° 85° $ 11. mC # mD 1 mE % 13. mF & A . A A + , 90° 90° 90° 90° mK mL mM mN 28 59° 73° 121° 107° Holt Geometry Name Date Class Name Practice A LESSON 11-4 Inscribed Angles 2. If inscribed angles of a circle intercept the same arc or are subtended by the same chord or arc, then the angles are congruent half 3. The measure of an inscribed angle is intercepted arc. Find each measure. supplementary 1. If a quadrilateral is inscribed in a circle, then its opposite angles are . � 1. . � the measure of its 5. � � � � ��� 30° 140° ��� 6. ��� � 15 8. 25 z� � � ��� � � 6. � 42° m�VUS � � 10. ��������� � � 71° m�ZWY � � � � ��������� � � 13 y� 9. 6 a� ���� � � m�B � m�C � m�D � m�E � � � 120° 90° 60° 90° � 12. ��� ��� ���� � � � 130° 100° 50° 80° m�F � m�G � m�H � m�I � ������������ 10. m�X � � ���������� � m�W � m�T � m�U � ���������� � ���� ��� � 12. � � m�V � �� � � ��� � m�W � � 27 Name Date m�Z � ��������� � Copyright © by Holt, Rinehart and Winston. All rights reserved. m�Y � ���������� � � 105° Class Holt Geometry 71° 109° 109° 71° � 11. m�D � m�E � � 68° 95° 112° 85° 13. m�F � � ��������� � ��������� ��������� � � m�L � m�M � m�N � 28 Copyright © by Holt, Rinehart and Winston. All rights reserved. Name Date Class Holt Geometry Inscribed Angle Theorem _ The measure of an inscribed angle is half the measure of its intercepted arc. � _ _ Possible answer: It is given that AC � AD. In a circle, congruent � � � � �ABC is an inscribed angle. � AC is an intercepted arc. � � chords intercept congruent arcs, so ABC � AED. DC is congruent to itself by � the Reflexive Property of Congruence. By the Arc Addition Postulate and the � � � m�ABC � _1_ m AC 2 � Addition Property of Congruence, ACD � ADC. �ABC intercepts ADC, so � � � m�ABC � _1_ mADC. �AED intercepts ACD, so m�AED � _1_ mACD. By 2 2 substitution, m�ABC � m�AED. Therefore �ABC � �AED. � � PQ RS _ �_ � � Inscribed Angles If inscribed angles of a circle intercept the same arc, then the angles are congruent. � �ABC and �ADC intercept AC, so �ABC � �ADC. � Possible answer: It is given that PQ � RS. By the definition of � � � � � congruent arcs, mPQ � mRS. �PSQ intercepts PQ, and �RQS � intercepts RS. So m�PSQ must equal m�RQS. Therefore �PSQ _ � �RQS. _ �PSQ _ and _�RQS are congruent alternate interior angles of QR and PS. So QR � PS. � An inscribed angle subtends a semicircle if and only if the angle is a right angle. � � � � � � � � Find each measure. � � 1. m�LMP and m MN For each quadrilateral described, tell whether it can be inscribed in a circle. If so, describe a method for doing so using a compass and straightedge, and draw an example. 2. m�GFJ and m FH � � � 3. a parallelogram that is not a rectangle or a square 110° cannot be inscribed in a circle � 4. a kite � Can be inscribed in a circle; possible answer: The two congruent angles of the kite are opposite, so they must be right angles. Draw a diameter. Draw segments from opposite ends of the diameter to any point on the circle. Use the compass to copy one of the segments across the diameter. Draw the fourth side. 36° � � 48° 36° � � � m�GFJ � 55°; m FH � 72° m�LMP � 18°; m MN � 96° Find each value. 3. x 5. a trapezoid Can be inscribed in a circle; possible answer: The pairs of base angles of a trapezoid inscribed in a circle must be congruent. Draw any inscribed angle. Use the compass to copy the arc that this angle intercepts. Mark off the same arc from the vertex of the inscribed angle. Connect the points. 4. m�FJH � � � (4� � 9)° � (5� � 8)° � 6. a rhombus that is not a square � � 5� ° � cannot be inscribed in a circle Copyright © by Holt, Rinehart and Winston. All rights reserved. 59° 73° 121° 107° m�K � � � � Write paragraph proofs for Exercises 1 and 2. 2. Given: Prove: QR � PS 90° 90° 90° 90° m�C � � � Reteach LESSON 11-4 Inscribed Angles Practice C LESSON 11-4 Inscribed Angles 1. Given: AC � AD Prove: �ABC � �AED � Find the angle measures of each inscribed quadrilateral. � 13. Iyla has not learned how to stop on ice skates yet, so she just skates straight across the circular rink until she hits a wall. She starts at P, turns 75° at Q, and turns 100° at R. Find how many degrees Iyla will turn at S to get back to her starting point. 77° m�SRT � ��������� � � Find the angle measures of each inscribed quadrilateral. � � � � � _ � 7. ���������� 48° � 8. ��������� ��� ���� � � ���������� � � � 73° � ��� � m�CED � 11. � � ��� ��� ������� � �� � � � � � x� � 9. 10° 90.5° m�XVU � Find each value. � � ���� � Find each value. 7. � 5. A circular radar screen in an air traffic control tower shows these flight paths. Find m�LNK. � � m�VXW � � � mGH � ��� 4. � ��� � 45° 40° 130° 138° � mTSR � 9° 78° m�FGI � � � � mQRS � � m�IHJ � � mGH � � 2. ��� � � ��� m�BAC � � mFE � � mDEA � � � Find each measure. � � � 3. 33° 192° m�CED � ��� ��� 4. An inscribed angle subtends a semicircle if and only if the angle is a right angle . ��� Class Practice B LESSON 11-4 Inscribed Angles In Exercises 1–4, fill in the blanks to complete each theorem. � Date 45° 16.4 29 Copyright © by Holt, Rinehart and Winston. All rights reserved. Holt Geometry Copyright © by Holt, Rinehart and Winston. All rights reserved. 65 30 Holt Geometry Holt Geometry