10.5. Angle Relationships in Circles For use with Exploration Essential Question When a chord intersects a tangent line or another

Transcription

1 Name ate 0.5 ngle Relationships in ircles For use with Exploration 0.5 Essential Question When a chord intersects a tangent line or another chord, what relationships exist among the angles and arcs formed? EXPLORTION: ngles Formed by a hord and Tangent Line Go to igideasmath.com for an interactive tool to investigate this exploration. Work with a partner. Use dynamic geometry software. Sample a. onstruct a chord in a circle. t one of the endpoints of the chord, construct a tangent line to the circle. b. Find the measures of the two angles formed by the chord and the tangent line. 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. ngle Measure ngle Measure ircular rc Measure ircular rc Measure 98 Geometry opyright ig Ideas Learning, LL ll rights reserved.

2 Name ate 0.5 ngle Relationships in ircles (continued) EXPLORTION: ngles Formed by Intersecting hords Go to igideasmath.com for an interactive tool to investigate this exploration. Work with a partner. Use dynamic geometry software. Sample a. onstruct two chords that intersect inside a circle. b. Find the measure of one of the angles formed by the intersecting chords. c. Find the measures of the arcs intercepted by the angle in part (b) and its vertical angle. What do you observe? E d. Repeat parts (a) (c) several times. Record your results in the following table. Then write a conjecture that summarizes the data. ngle Measure rc Measures Observations ommunicate Your nswer 3. When a chord intersects a tangent line or another chord, what relationships exist among the angles and arcs formed? 48 m 4. Line m is tangent to the circle in the figure at the right. Find the measure of. 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. opyright ig Ideas Learning, LL ll rights reserved. Geometry 99

3 Name ate 0.5 Notetaking with Vocabulary For use after Lesson 0.5 In your own words, write the meaning of each vocabulary term. circumscribed angle Theorem 0.4 Tangent and Intersected hord 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. m = m m = m ore oncepts Intersecting Lines and ircles 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 Theorem 0.5 ngles Inside the ircle 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. m = (m + m ), m = (m + m ) 300 Geometry opyright ig Ideas Learning, LL ll rights reserved.

4 Name ate 0.5 Notetaking with Vocabulary (continued) Theorem 0.6 ngles Outside the ircle 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. m = (m m ) P R Q m = (mpqr mpr ) W 3 Z X Y m 3 = (mxy mwz ) ore oncepts ircumscribed ngle circumscribed angle is an angle whose sides are tangent to a circle. circumscribed angle Theorem 0.7 ircumscribed ngle Theorem The measure of a circumscribed angle is equal to 80 minus the measure of the central angle that intercepts the same arc. m = 80 m opyright ig Ideas Learning, LL ll rights reserved. Geometry 30

5 Name ate 0.5 Notetaking with Vocabulary (continued) Extra Practice In Exercises 3, is tangent to the circle. Find the indicated measure.. m. m me E In Exercises 4 and 5, m = 0 and m =. Find the indicated measure. 4. m 5. m In Exercises 6 9, find the value of x x (5x + 4) P x (4x + 8) J Q 38 R 7 S L (3x + ) 8 K 30 Geometry opyright ig Ideas Learning, LL ll rights reserved.