# 2 Explain 1 Proving the Intersecting Chords Angle Measure Theorem

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1 xplain 1 Proving the Intersecting hords ngle easure Theorem In the xplore section, you discovered the effects that line segments, such as chords and secants, have on angle measures and their intercepted arcs. These relationships can be stated as theorems, with the first one about chords. The Intersecting hords ngle easure Theorem If two secants or chords intersect in the interior of a circle, then the measure of each angle formed is half the sum of the measures of its intercepted arcs. _ hords and _ intersect at. m 1 (m + m ) 1 Houghton ifflin Harcourt Publishing ompany odule Lesson 5

2 xample 1 Prove the Intersecting hords ngle easure Theorem Given: and intersect at. Prove: m 1 (m + m ) 1 Statements Reasons 1. _ and _ intersect at. 1. Given _. raw.. Through any two points, there is exactly one line. 3. m 1 m + m 3. xterior ngle Theorem 4. m _ m, 4. Inscribed ngle Theorem m _ m 5. m 1 _ m + _ m 5. Substitution Property 6. m 1 _ 6. istributive Property (m + m ) Reflect 3. Iscusssion xplain how an auxiliary segment and the xterior ngle Theorem are used in the proof of the Intersecting hords ngle easure Theorem. 1 is formed by two intersecting chords. n auxiliary line is drawn to create a triangle using parts of the chords as the other two sides. Then, 1 is an exterior angle for the triangle formed, and its measure is the sum of the measures of its remote interior angles. Those interior angles intercept the same arcs as 1, so its measure can be found. Houghton ifflin Harcourt Publishing ompany Find each unknown measure. 4. m PK 5. m PR 111 K P L 61 m PK (m K + m L ) ( ) (17 ) 86 R P T 58 8 m TS (m S + m PR ); 58 (8 + m PR ); 116 (8 + m PR ); m PR ; 34 m PR S odule Lesson 5

3 xplain pplying the Tangent-Secant Interior ngle easure Theorem The angle and arc formed by a tangent and secant intersecting on a circle also have a special relationship. The Tangent-Secant Interior ngle easure Theorem If a tangent and a secant (or a chord) intersect on a circle at the point of tangency, then the measure of the angle formed is half the measure of its intercepted arc. Tangent and secant intersect at. m m xample Find each unknown measure. m m m (14 ) 71 m ( m m ( m ) 90 ) 180 m 14 Find the measure. 6. m P 7. m P m P ( m P ) 61 ( m P ) 1 m P xplain 3 m P m P (38 ) pplying the Tangent-Secant xterior ngle easure Theorem You can use the difference in arc measures to find measures of angles formed by tangents and secants intersecting outside a circle. 119 P Houghton ifflin Harcourt Publishing ompany odule Lesson 5

4 The Tangent-Secant xterior ngle easure Theorem If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is half the difference of the measures of its intercepted arcs. 1 F G H L K 3 J m 1 (m - m ) m (m HG - m G ) m 3 (m J - m K ) xample 3 Find the value of x. J 5 K L x 83 m L (m J - m K ) 5 (83 - x ) x x 33 x Houghton ifflin Harcourt Publishing ompany x Find the value of x. 8. x x x x - ( ) (38 ) (38-1 ( )) 116 ( ) x 58 x ( ) x (94) x 47 odule Lesson 5

5 9. The superior oblique and inferior oblique are two muscles that help control eye movement. They intersect behind the eye to create an angle, as shown. If m 5, what is m? x (5 - (360-5 ) ) x (5-135) x (90) x 45 Superior oblique Inferior oblique So, the measure of the angle between the two muscles is 45. xplain 4 Understanding ngle Relationships in ircles You can summarize angle relationships in circles by looking at where the vertex of the angle lies: on the circle, inside the circle, or outside the circle. ngle Relationships in ircles Vertex of the ngle On a circle easure of ngle Half the measure of its intercepted arc iagrams m 1 60 m 100 Inside a circle Outside a circle Half the sum of the measures of its intercepted arcs Half the difference of the measures of its intercepted arcs m 1 ( ) m 1 (0-78 ) m (15-45 ) 6 40 Houghton ifflin Harcourt Publishing ompany odule Lesson 5

6 xample 4 Find the unknown arc measures F 50 Find m F. m ( m F - m F ) 50 ( m F ) 100 ( 50 - m F ) -50 -m F 50 m F Find m. m - (m + m F + mf ) ( ) Find m K. J 48 K L P m P (m JK + m P ) 79 (48 + m P ) 110 m P m K (m JK + m JP + m P ) ( ) laborate 116 Houghton ifflin Harcourt Publishing ompany 11. omplete the graphic organizer that shows the relationship between the angle measurement and the location of its vertex. On the ircle The angle measure is half the measure of its intercepted arc. ngle s Vertex Location Inside the ircle The angle measure is half the sum of the measures of its intercepted arcs. Outside the ircle The angle measure is half the difference of the measures of its intercepted arcs. 1. ssential uestion heck-in What is similar about all the relationships between angle measures and their intercepted arcs? The arc measurements have to be halved to obtain the angle measurement. odule Lesson 5

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