1. Bellwork  One-half of the measure of an angle plus its supplement is equal to the measure of the angle. Find the measure of the angle  Solve for x 2x 2 -13x-7=0 Clickers

2. Bellwork Solution  One-half of the measure of an angle plus its supplement is equal to the measure of the angle. Find the measure of the angle  Solve for x 2x 2 -13x-7=0

3. Section 10.5

4. The Concept  For the bulk of this chapter we’ve been dealing with the arc of circles  Today we’re going to finish out our discussion of arcs and angles

5. Theorem Theorem 10.11 If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc A C B D This makes sense if we think about this in terms of an inscribed angle with only one leg

6. On your own  What is the measure of arc ADC 82 o A C B D

7. On your own  What is the measure of arc ABC 82 o A C B D

8. On your own  What is the measure of angle ACB, if arc CD measures 160 o 35 o A C B D

9. Theorems Theorem 10.12 If two chords intersect inside a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle A B C D x

10. Example Find the measure of angle AEC A B C D E 110 o 96 o

11. On your own  What is the measure of angle ABC 225 o A C B D E 255 o

12. On your own  What is the measure of angle ABD 148 o A C B D E 30 o

13. On your own  What is the measure of arc AC 121 o A C B D E 142 o

14. Theorems Theorem 10.13 Angles Outside the Circle Theorem If a tangent and a secant, two tangents or two secants intersect outside a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs A B C D

15. Theorems Find the measure of angle ADB A B D 250 o 110 o C

16. On your own  What is the measure of angle ABD 30 o A C B D E 86 o

17. On your own  What is the measure of angle ADB 65 o A B D 195 o

18. On your own  What is the measure of arc BD 32 o A B D 164 o

19. On your own  What is the measure of arc AB 22 o A B D 120 o

20. On your own  What is the measure of angle of E, if arc ABC is 200 o A B D C E

21. On your own  What is the measure of angle of F? A B D C F 110 o 80 o

22. On your own  What is the measure of angle of F? A B D C F 35 o 110 o 105 o

23. Homework  10.5 1-20, 22-25

24. Most Important Points  Angle-Chord Relationships that intercept inside a circle  Angle relationship that intersects outside the circle