1. For each circle C, find the value of x. Assume that segments that appear to be tangent are tangent. 1)2)

2. Inscribed Angle: An angle whose vertex is on the circle and whose sides are chords of the circle INSCRIBED ANGLE INTERCEPTED ARC

3. Determine whether each angle is an inscribed angle. Name the intercepted arc for the angle. C L O T 1. YES; CL

4. Determine whether each angle is an inscribed angle. Name the intercepted arc for the angle. Q R K V 2. NO; QVR S

5. 160 0 80 0 To find the measure of an inscribed angle…

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?How do we solve for y? The measure of the inscribed angle is HALF the measure of the inscribed arc!!

7. Examples 3. If m JK = 80 , find m <JMK. M Q K S J 4. If m <MKS = 56 , find m MS. 40  112 

8. 72  If two inscribed angles intercept the same arc, then they are congruent.

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

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

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

12. z y 110 85 110 + y =180 y = 70 z + 85 = 180 z = 95 Example 8 Find y and z.

13. 180  diameter If a right triangle is inscribed in a circle then the hypotenuse is the diameter of the circle.

14. H K G N 4x – 14 = 90 Example 6 In  K, m <GNH = 4x – 14. Find the value of x. x = 26

15. H K G N or 6x – 5 + 3x – 4 = 90 Example 7 In  K, m <1 = 6x – 5 and m <2 = 3x – 4. Find the value of x. x = 11 1 2 6x – 5 + 3x – 4 + 90 = 180