1. Warm up

2. P A B Case I: Central Angle: Vertex is AT the center 

3. Case II: Inscribed Angle: Vertex is ON circle ANGLE ARC ANGLE ARC  

4. Intercepted Arc: An arc whose endpoints are the two points of intersection of an angle with the circle and all points that lie within the angle.

5. The arc is twice as big as the angle!!

6. 120  x y Find the value of x and y   

7. Examples 1. If m JK = 80  and  JMK = 2x – 4, find x. M Q K S J 2. If m  MKS = 56 , find m MS. x = 22 112 

8. 72˚ If two inscribed angles intercept the same arc, then they are congruent. Find the measure of  DOG and  DIG D O G I

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

10. Circumscribed Circle The circumscribed circle (or circumcircle) of a polygon is a circle which passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.

11. a quadrilateral inscribed in a circle: opposite angles are supplementary. A B C D

12. If a right triangle is inscribed in a circle then the hypotenuse is the diameter of the circle. diameter

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

14. 4x – 14 = 90 H K G N Example 4 In  K, GH is a diameter and m  GNH = 4x – 14. Find the value of x. x = 26 Bonus: What type of triangle is this? Why?

15. z y 110 85 110 + y =180 y = 70 z + 85 = 180 z = 95 Example 5 Find y and z.