1. Geometry 10.4 Other Angle Relationships in Circles

2. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles2 Goal  Use angles formed by tangents, secants, and chords to solve problems.

3. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles3 Review Note: in solving an equation with fractions, one of the first things to do is always “clear the fractions”.

4. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles4 You do it. Solve:

5. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles5 Review The measure of an inscribed angle is equal to one-half the measure of the intercepted arc. 8040 What if one side of the angle is tangent to the circle?

6. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles6 Theorem 10.2: Tangent-Chord A B C 12 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 the intercepted arc.

7. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles7 Simplified Formula aa bb 1 2

8. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles8 Example 1 A B C 80 160  200 

9. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles9 Example 2. Solve for x. A B C 4x (10x – 60)

10. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles10 If two lines intersect a circle, where can the lines intersect each other? On the circle. Inside the circle. Outside the circle. We already know how to do this.

11. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles11 Theorem 10.13 (Inside the circle) A B C D 1 If two chords intersect in a circle, then the measure of the angle is one-half the sum of the intercepted arcs.

12. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles12 Simplified Formula 1 aa bb

13. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles13 Example 3Find m1. A B C D 1 30 80

14. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles14 Example 4Solve for x. A B C D 60 20 xx 100 Check: 100 + 20 = 120 120 ÷ 2 = 60 

15. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles15 Your turn. Solve for x & y. A B C D xx 75 85 M yy K P O 20 32

16. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles16

17. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles17 Secant-Secant C A B D 1

18. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles18 Simplified Formula 1bb aa

19. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles19 Secant-Tangent C A B 1

20. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles20 Simplified Formula aa bb 1

21. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles21 Tangent-Tangent A B 1 C

22. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles22 Simplified Formula 1 aa bb

23. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles23 Example 5Find m1. 1 80 1035

24. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles24 Example 6Find m1. 1 120 70 25

25. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles25 Example 7Find m1. 1 210 150 30 ? 360 – 210 = 150 k m Rays k and m are tangent to the circle.

26. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles26 How to remember this:  If the angle vertex is on the circle, its measure is one-half the intercepted arc.  If an angle vertex is inside the circle, its measure if one-half the sum of the intercepted arcs.  If an angle vertex is outside the circle, its measure is one-half the difference of the intercepted arcs.