1. Inscribed Angles

2. Inscribed Angle: An angle that has its vertex on the circle and its sides contained in chords of the circle
Vertex B is on the circle B
Arc ADC is the arc intercepted by angle ABC.
AB and BC are chords of the circle A C

3. Theorem 10.5 Inscribed Angle Theorem
If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc (or the measure of the intercepted arc is twice the measure of the inscribed angle.

4. Example 1 p. 579
Use circle O on pg 579 & review the example shown. mAB = 140,mBC=100, mAD=mDC. Find the measures of angle 4 & 5.
mAB = 140 therefore, measure of angle 4 is ½ of Measure of angle 4 = 70
mBC = 100 therefore, measure of angle 5 is ½ of Measure of angle 5 = 50

5. Theorem 10.6: If two inscribed angles of a circle (or congruent circles) intercept congruent arcs or the same arc, then the angles are congruent. B A B C A D F D C E

6. Review the Proof in Ex. 2 p. 580 Try check your progress #2
Answer: Statements (Reasons)
1) RT bisects SU (Given)
2) SV = VU (def of segment bisector)
3) Angle SRT intercepts arc ST. Angle SUT intercepts arc ST. (def of intercepted arc)
4) Angle SRT = angle SUT (inscribed angles of same arc are congruent)
5) Angle RVS = angle UVT (vertical angles are congruent
6) Triangle RVS = Triangle UVT (AAS)

7. Theorem 10.7
If the inscribed angle of a triangle intercepts a semicircle, the angle is a right angle. A D
Arc ADC is a semicircle, so the measure of angle ABC is 90. B C

8. Refer to Circle F & given info in Ex. 4 on pg.581
Find the measure of angle 3 & angle 4.
Answer
Since Arc AD = Arc BD, then angle 3 & angle 4 are also equal. Therefore, each are 45 since Arc ADE is a semicircle so angle B is 90 leaving the other two angles (3 & 4) are complementary (add to 90).

9. Refer to Circle V on pg. 582
Quadrilateral WXYZ is inscribed in circle V. If the measure of angle W = 95 and measure of angle Z is 60, find the measure of angle X and Y.
Arc WXY = 120 & Arc ZYX = 190
This means that arc WZY = = 240 & XWZ = = 170
Angle Y is ½ of 170 = 85
Angle X is ½ of 240 = 120

10. Homework #66
Study Guide 10.4 Worksheet