1. Warm-up Find the measure of each arc.

2. Inscribed Angles

3. Inscribed Angle: An angle whose vertex is on the circle INSCRIBED ANGLE INTERCEPTED ARC

4. Can also look like this:

5. 160  80 

6. 120 x What do we call this type of angle? What is the value of x? y What do we call this type of angle?

7. Find m  1. 84° 1

8. Find m  1. 108° 1

9. Examples If m JK = 80 , find m  JMK. M Q K S J If m  MKS = 56 , find m MS. 40  112 

10. 72˚ If two inscribed angles intercept the same arc, then they are congruent. Find the measure of angle DOG and angle DIG D O G I

11. In  J, m  3 = 5x and m  4 = 2x + 9. Find the value of x. 3 Q D J T U 4 x = 3

12. If all the vertices of a polygon touch the edge of the circle, the polygon is INSCRIBED and the circle is CIRCUMSCRIBED.

13. A circle can be circumscribed around a quadrilateral if and only if its opposite angles are supplementary. A B C D

14. Proof of previous theorem

15. z y 110 85 110 + y =180 y = 70 z + 85 = 180 z = 95 Find y and z.

16. H K G N 4x – 14 = 90 In  K, GH is a diameter and m  GNH = 4x – 14. Find the value of x. x = 26

17. H K G N 6x – 5 + 3x – 4 = 90 In  K, m  1 = 6x – 5 and m  2 = 3x – 4. Find the value of x. x = 11 1 2