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Name _________________________________________________________ Date _________ Relationships in Circles 10.5 Angle For use with Exploration 10.5 Essential Question When a chord intersects a tangent line or another chord, what relationships exist among the angles and arcs formed? 1 EXPLORATION: Angles Formed by a Chord and Tangent Line Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Use dynamic geometry software. Sample a. Construct a chord in a circle. At one of the endpoints of the chord, construct a tangent line to the circle. B C b. Find the measures of the two A angles formed by the chord and the tangent line. D c. Find the measures of the two circular arcs determined by the chord. d. Repeat parts (a)–(c) several times. Record your results in the following table. Then write a conjecture that summarizes the data. Angle Measure 1 298 Geometry Student Journal Angle Measure 2 Circular Arc Measure 1 Circular Arc Measure 2 Copyright © Big Ideas Learning, LLC All rights reserved. Name_________________________________________________________ 10.5 2 Date __________ Angle Relationships in Circles (continued) EXPLORATION: Angles Formed by Intersecting Chords Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Use dynamic geometry software. Sample a. Construct two chords that intersect inside a circle. D b. Find the measure of one of the angles formed by B the intersecting chords. c. Find the measures of the arcs intercepted by the C angle in part (b) and its vertical angle. What do you observe? A E d. Repeat parts (a)–(c) several times. Record your results in the following table. Then write a conjecture that summarizes the data. Angle Measure Arc Measures Observations Communicate Your Answer 3. When a chord intersects a tangent line or another chord, what 148° relationships exist among the angles and arcs formed? 1 m 4. Line m is tangent to the circle in the figure at the right. Find the measure of ∠1. 5. Two chords intersect inside a circle to form a pair of vertical angles with measures of 55°. Find the sum of the measures of the arcs intercepted by the two angles. Copyright © Big Ideas Learning, LLC All rights reserved. Geometry Student Journal 299 Name _________________________________________________________ Date _________ with Vocabulary 10.5 Notetaking For use after Lesson 10.5 In your own words, write the meaning of each vocabulary term. circumscribed angle Theorems Theorem 10.14 Tangent and Intersected Chord Theorem If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one-half the measure of its intercepted arc. B C 2 1 A m∠1 = 12 mAB m∠2 = 12 mBCA Notes: Core Concepts Intersecting Lines and Circles 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 Notes: Theorems D Theorem 10.15 Angles Inside the Circle Theorem If two chords intersect inside a circle, then the measure of each angle is one-half the sum of the measure of the arcs intercepted by the angle and its vertical angle. Notes: 300 Geometry Student Journal A 1 B 2 C m∠1 = 12 (mDC + mAB ), m∠2 = 12 (mAD + mBC ) Copyright © Big Ideas Learning, LLC All rights reserved. Name_________________________________________________________ 10.5 Date __________ Notetaking with Vocabulary (continued) Theorem 10.16 Angles Outside the Circle Theorem If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is one-half the difference of the measures of the intercepted arcs. B P A 1 W 3 Z 2 C m∠1 = X Q 1 (mBC 2 R − mAC ) m∠2 = 1 (mPQR 2 − mPR ) m∠3 = Y 1 (mXY 2 − mWZ ) Notes: Core Concepts Circumscribed Angle A circumscribed angle is an angle whose sides are tangent to a circle. Notes: A B circumscribed C angle Theorems Theorem 10.17 Circumscribed Angle Theorem The measure of a circumscribed angle is equal to 180° minus the measure of the central angle that intercepts the same arc. Notes: A D C B m∠ADB = 180° − m∠ACB Copyright © Big Ideas Learning, LLC All rights reserved. Geometry Student Journal 301 Name _________________________________________________________ Date _________ 10.5 Notetaking with Vocabulary (continued) Extra Practice In Exercises 1–3, CD is tangent to the circle. Find the indicated measure. A 2. m AB 1. m∠ABC D 70° 3. m AEB E B C = 220° and m∠B = 21°. In Exercises 4 and 5, mADB Find the indicated measure. 4. m AB A C D 5. m∠ACB B In Exercises 6–9, find the value of x. 6. C 7. 50° D C 97° D x° B A (5x + 14)° 92° A 8. B 35° 9. P (4x + 8)° x° J 72° Q 38° R S L (3x + 2)° 28° K 302 Geometry Student Journal Copyright © Big Ideas Learning, LLC All rights reserved.