1. 10.4 Other Angle Relationships in Circles
Geometry
10.4 Other Angle Relationships in Circles

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

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

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

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

6. Theorem 10.2: Tangent-Chord
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. B C 2 1 A
Thursday, March 23, 2:46
Geometry 10.4 Other Angle Relationships in Circles

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

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

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

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

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

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

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

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

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

16. Geometry 10.4 Other Angle Relationships in Circles
Intersection
Outside the
Circle
Thursday, March 23, 2:46
Geometry 10.4 Other Angle Relationships in Circles

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

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

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

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

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

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

23. Intersection Outside the Circle
Secant-Secant
Secant-Tangent
Tangent-Tangent
In all cases, the measure of the exterior angle is found the same way: One-half the difference of the larger and smaller arcs.
(Click the titles above for Sketchpad Demonstrations.)
Thursday, March 23, 2:46
Geometry 10.4 Other Angle Relationships in Circles

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

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

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

27. Geometry 10.4 Other Angle Relationships in Circles
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.
Thursday, March 23, 2:46
Geometry 10.4 Other Angle Relationships in Circles

28. Geometry 10.4 Other Angle Relationships in Circles
What can you do?
Open a book to page 624.
Do problems 2 – 7.
Carefully consider what the situation is and use the correct formula.
6 minutes.
Thursday, March 23, 2:46
Geometry 10.4 Other Angle Relationships in Circles

29. Geometry 10.4 Other Angle Relationships in Circles
Answers
210 [2  105]
60 [½( )]
65 [ ½(190 – 60)]
90 [ ½(270 – 90)]
22.5 [ ½(80 – 35)]
88 [ ½( )]
Thursday, March 23, 2:46
Geometry 10.4 Other Angle Relationships in Circles

30. Geometry 10.4 Other Angle Relationships in Circles
Homework
Thursday, March 23, 2:46
Geometry 10.4 Other Angle Relationships in Circles