15.5 Angle Relationships in Circles

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1 ame lass ate 15.5 ngle Relationships in ircles ssential uestion: What are the relationships between angles formed by lines that intersect a circle? xplore xploring ngle Measures in ircles The sundial is one of many instruments that use angles created in circles for practical applications, such as telling time. Resource Locker In this lesson, you will observe the relationships between angles created by various line segments and their intercepted arcs. Using geometry software, construct a circle with two secants and F that intersect inside the circle at G, as shown in the figure. Houghton Mifflin Harcourt Publishing ompany onomacs/istockphoto. com reate two new points H and I that are on the circle as shown. These will be used to measure the arcs. Hide if desired. Measure GF formed by the secant lines, and measure H and IF. Record angle and arc measurements in the first column of the table. m GF m H m IF Sum of rc Measures rag F around the circle and record the changes in measures in the table in Part. Try to create acute, right, and obtuse angles. e sure to keep H between and and I between and F for accurate arc measurement. Move them if necessary. Module Lesson 5

2 Reflect 1. an you make a conjecture about the relationship between the angle measure and the two arc measures?. Using the same circle you created in step, drag points around the circle so that the intersection is outside the circle, as shown. Measure FG formed by the secant lines and measure IF and H. rag points around the circle and observe the changes in measures. Record some measures in the table. m FG m IF m H ifference of rc Measures What is similar and different about the relationships between the angle measure and the arc measures when the secants intersect outside the circle? xplain 1 Proving the Intersecting hords ngle Measure 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 Measure 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 Mifflin Harcourt Publishing ompany Module Lesson 5

3 xample 1 Prove the Intersecting hords ngle Measure Theorem Given: and intersect at. Prove: m 1 = (m + m ) 1 Statements 1. _ and _ intersect at. 1. Given Reasons. raw _.. Through any two points, there is exactly one line. 3. m 1 = m + m m = _ m, 4. m = _ m 5. m 1 = _ m + _ m 5. Substitution Property Reflect 3. Iscusssion xplain how an auxiliary segment and the xterior ngle Theorem are used in the proof of the Intersecting hords ngle Measure Theorem. Houghton Mifflin Harcourt Publishing ompany Your Turn Find each unknown measure. 4. m MPK 5. m PR K 61 M P L 111 R P T 58 8 S Module Lesson 5

4 xplain pplying the Tangent-Secant Interior ngle Measure 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 Measure 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 ) = ) = m 14 Your Turn Find the measure. 6. m P 7. m MP xplain 3 pplying the Tangent-Secant xterior ngle Measure Theorem You can use the difference in arc measures to find measures of angles formed by tangents and secants intersecting outside a circle. P M Houghton Mifflin Harcourt Publishing ompany Module Lesson 5

5 The Tangent-Secant xterior ngle Measure 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 M J m 1 = ( m - m ) m = ( m HG - m G ) m 3 = (m J - m KM ) xample 3 Find the value of x. J 5 K L x 83 M m L = (m J - m 5 = 50 = 83 - x -33 = - x (83 - x ) KM ) 33 = x Houghton Mifflin Harcourt Publishing ompany x Your Turn Find the value of x. 8. x x = x = x = x = (38 - (360 - ) ) ( (38 - ) ) ( ) Module Lesson 5

6 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? Superior oblique Inferior oblique 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 Measure 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 Mifflin Harcourt Publishing ompany Module Lesson 5

7 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 ) = - ( ) = - = Your Turn 10. Find m K. M 48 J K P L laborate Houghton Mifflin Harcourt Publishing ompany 11. omplete the graphic organizer that shows the relationship between the angle measurement and the location of its vertex. The angle measure is half the measure of its intercepted arc. ngle s Vertex Location Inside the ircle Outside the ircle 1. ssential uestion heck-in What is similar about all the relationships between angle measures and their intercepted arcs? Module Lesson 5

8 valuate: Homework and Practice Use the figure for xercises 1. Suppose you use geometry software to construct a secant and tangent that intersect on a circle at point. Online Homework Hints and Help xtra Practice 1. Suppose you measure and you measure. Then you drag the points around the circle and measure the angle and arc three more times. What would you expect to find each time? Which theorem from the lesson would you be demonstrating?. When the measure of the intercepted arc is 180, what is the measure of the angle? What does that tell you about the secant? Find each measure. 3. m PR 4. m S P T R 5. m MKJ J 6. m PK 38.5 K M Houghton Mifflin Harcourt Publishing ompany L 51.5 K P 61 M L Module Lesson 5

9 Find each measure. Use the figure for xercises m 8. m 11 Find each measure. Use the figure for xercises x Y 40 V 180 U Z W 9. m XZW 10. m YXZ Find the value of x x 140 x 170 Houghton Mifflin Harcourt Publishing ompany 13. x Module Lesson 5

10 14. Represent Real-World Problems Stonehenge is a circular arrangement of massive stones near Salisbury, ngland. viewer at V observes the monument from a point where two of the stones and are aligned with stones at the endpoints of a diameter of the circular shape. Given that m = 48, what is m V? V 15. Multi-Step Find each measure. a. Find m P. M J 48 K L P b. Use your answer to part a to find m K. 16. Multi-Step Find each measure. a. Find m. b. Use your answer to part a to find m F J G H 61 F Houghton Mifflin Harcourt Publishing ompany Jason Hawkes/orbis Module Lesson 5

11 MS P and m PS = 50. Find each measure. 17. m PR 50 mlr = 170 M S L R P 18. m LP 19. Represent Real-World Problems satellite orbits Mars. When it reaches S it is about 1,000 km above the planet. What is x, the measure of the arc that is visible to a camera in the satellite? S x 38 Houghton Mifflin Harcourt Publishing ompany StockTrek/Photodisc/ Getty Images 0. Use the circle with center J. Match each angle or arc on the left with its measure on the right. Indicate a match by writing the letter for the angle or arc on the line in front of the corresponding measure F J F Module Lesson 5

12 1. Use the Plan for Proof to write a proof for one case of the Tangent-Secant xterior ngle Measure Theorem. 1 Given: Tangent and secant Prove: m = ( m - m ) Plan: raw auxiliary line segment _. Use the xterior ngle Theorem to show that m = m - m. Then use the Inscribed ngle Theorem and the Tangent-Secant Interior ngle Measure Theorem.. Justify Reasoning Write a proof that the figure shown is a square. Given: _ YZ and _ WZ are tangent to circle X, m WY = 90 Prove: WXYZ is a square. X Y W Z Houghton Mifflin Harcourt Publishing ompany Module Lesson 5

13 H.O.T. Focus on Higher Order Thinking 3. Justify Reasoning Prove the Tangent-Secant Interior ngle Theorem. Given: Tangent and secant Prove: m = m (Hint: onsider two cases, one where _ is a diameter and one where _ is not a diameter.) Houghton Mifflin Harcourt Publishing ompany 4. ritical Thinking Suppose two secants intersect in the exterior of a circle as shown. Which is greater, m 1 or m? Justify your answer. 1 Module Lesson 5

14 Lesson Performance Task The diameter of the Moon is about 160 miles. From arth, the portion of the Moon s surface that an observer can see is from a circumscribed angle of approximately a. Find the measure of. xplain how you found the measure. b. What fraction of the circumference of the Moon is represented by? c. Find the length of. You can use the formula = πr to find the circumference of the Moon. Houghton Mifflin Harcourt Publishing ompany Module Lesson 5

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