1. Warm Up Section 4.5 Find x: 1. 2. 3. 4. 5. 6. xoxo xoxo 70 o 32 o xoxo xoxo 100 o x 12 xoxo 45 o

2. Answers to Warm Up Section 4.5 Find x: 1. 2. 3. 4. 5. 6. xoxo xoxo 70 o 32 o xoxo xoxo 100 o x 12 xoxo 45 o 90 o 140 o 32 o 40 o 45 o

3. Angles Formed by Chords, Secants, and Tangents Section 4.5 Standard: MM2G3 bd Essential Question: How are properties of chords, tangents, and secants used to find angle measures?

4. Recall how arcs are related to central angles: arc = angle Ex. x = 65

5. Recall how an arc is related to its inscribed angle: angle = ½ arc Ex. arc = angle × 2 angle = arc ÷ 2 x = 80 x = 32

6. The measure of the angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs. 1 X Y Z W angle = ½ (arc 1 + arc 2)

7. Example 1: If = 45 o and = 75 o, find m S 1 R P Q m  1 = ½ (mRS + mPQ) m  1 = ½(75 o + 45 o ) m  1 = ½(120 o ) m  1 = 60 o 75 o 45 o

8. If and 80 o, find m  1 = ½ (mRS + mPQ) 55 o = ½(80 o + x) 110 o = (80 o + x) 30 o = x Example 2: S R P Q 80 o 55 o x

9. x°x° 80° 20° 100° x°x° Try these with your partner: 3. 4. x o = ½(80 o + 20 o ) x o = ½(100 o ) x o = 50 o 90 o = ½(100 o + x o ) 180 o = (100 o + x o ) 80 o = x o

10. 70° x°x° 40° 5. y o = ½(70 o + 40 o ) y o = ½(110 o ) y o = 55 o x o = 180 o – 55 o = 125 o yoyo

11. Case 2: Vertex outside the circle. The measure of an angle formed by 2 secants, 2 tangents, or a secant and a tangent is half the difference of the measures of the intercepted arcs. angle = ½ (arc 1 – arc 2)

12. A C B D E Example 6: If mDC = 100 o and mEB = 40 o, find m  A. 100 o 40 o m  A = ½ (mDC – mEB) x o = ½(100 o – 40 o ) x o = ½ (60 o ) x o = 30 o xoxo

13. W V X Y Z Example 7: If m  W = 65 o and mXZ = 70 o, find mXVY m  W = ½ (mXVY – mXZ) 65 o = ½(x o – 70 o ) 130 o = x o – 70 o 200 o = x o 70 o 65 o xoxo

14. P Q R S Example 8: If mQRS = 240 o, find mQS and m  P. m  P = ½ (mQRS – mQS) x o = ½(240 o – 120 o ) x o = ½ (120 o ) x o = 60 o 240 o mQS = 360 o – 240 o = 120 o 120 o xoxo

15. x°x° 230° 80° 140° x° 120° Try these with your partner: 9. 10. x o = ½(80 o – 20 o ) x o = ½ (60 o ) x o = 30 o x o = ½(230 o – 130 o ) x o = ½ (100 o ) x o = 50 o 130 o 20 o

16. x° 150° 35° 30° x° 100° 11. 12. x o = ½(50 o – 30 o ) x o = ½ (20 o ) x o = 10 o 35 o = ½(150 o – x o ) 70 o = (150 o – x o ) -80 o = – x o 80 o = x o 50 o

17. 1 x°x° The vertex of the angle is located at the center of the circle. So, the angle is a central angle and is equal to the measure of the intercepted arc. m  1 = x o Summary: Measures of Angles Formed by Radii, Chords, Tangents and Secants angle = arc

18. 2 x°x° 2 x°x° The vertex of the angle is a point on the circle. So, the measure of the angle is one half the measure of the intercepted arc. angle = ½arc m  2 = ½ x o

19. 3 x°x° y°y° The vertex of the angle is located in the interior of the circle and not at the center, so the measure of the angle is half the sum of the intercepted arcs. angle = ½(arc1 + arc2) m  3 = ½(x o + y o )

20. 4 x° y° 4 x° y° 4 x° y° The vertex of the angle is located in the exterior of the circle and not at the center, so the measure of the angle is half the difference of the intercepted arcs. angle = ½(arc1 – arc2) m  4 = ½(x o – y o )

21. 3 O B 4 1 2 9 6 7 8 5 10 A C D E BE is a diameter of the circle with center O. AT is tangent to the circle at A. mAB = 80 o, mBC = 20 o, and mDE = 50 o. 80 o 20 o 50 o 100 o 110 o

22. 3 O B 4 1 2 9 6 7 8 5 10 A C D E 80 o 20 o 50 o 100 o 110 o 13. m  1 = ½(80 o ) = 40 o 14. m  2 = ½(100 o ) = 50 o 15. m  3 = ½(80 o + 50 o ) = 65 o 16. m  4 = ½(100 o – 50 o ) = 25 o 17. m  5 = ½(80 o ) = 40 o 18. m  6 = ½(180 o ) = 90 o

23. 3 O B 4 1 2 9 6 7 8 5 10 A C D E 80 o 20 o 50 o 100 o 110 o 19. m  7 = ½(100 o ) = 50 o 20. m  8 = 20 o 21. m  9 = ½(50 o ) = 25 o 22. m  10 = ½(150 o ) = 75 o