1. Warm-Up
A circle and an angle are drawn in the same plane. Find all possible ways in which the circle and angle intersect at two points.

2. 9.4 – Angles Formed by Secants and Tangents
Topics:
Theorems about measures of arcs intercepted by angles

3. Classification of Angles with Circles
Case 1: Vertex is on the circle
Two secants
Secant and tangent

4. Classification of Angles with Circles
Case 2: Vertex is inside the circle
Two secants

5. Classification of Angles with Circles
Case 3: Vertex is outside the circle
Two secants
Secant and tangent
Two tangents

6. Vertex on Circle – Secant and Tangent

7. Vertex on Circle – Secant and Tangent
Part 1 – The secant-tangent angle is a right angle. P V C

8. Vertex on Circle – Secant and Tangent
Part 2 – The secant-tangent angle is acute. A P 3 1
Copy and complete the table. 2 V C

9. Vertex on Circle – Secant and Tangent
Part 3 – The secant-tangent angle is obtuse. P 1 3 A
Copy and complete the table. 2 V C

10. Theorem
If a tangent and a secant (or a chord) intersect on a circle at the point of tangency, then the measure of the angle formed is _______ the measure of its intercepted arc.
one-half

11. Vertex Inside Circle – Two Secants
Find the relationship between and the measures of A D 2 1 V P B
Copy and complete the table. C

12. Theorem
The measure of an angle formed by two secants or chords that intersect in the interior of a circle is _______ the _______ of the measures of the arcs intercepted by the angle and its vertical angle.
one-half
sum

13. Vertex Outside Circle – Two Secants
Find the relationship between and the measures of B 2 A 1 V C
Copy and complete the table. D

14. Theorem
The measure of an angle formed by two secants that intersect in the exterior of a circle is ________ the _________ of the measures of the intercepted arcs.
one-half
difference

15. Example 1 Given: AB is tangent to circle P at A, mAC = 60, mCD = 70,
and mDE = 80
Find mDAB, mCBA, and
mDGC.
Because the vertex of DAB is ON the circle, .
mDAB

16. Example 1 Given: AB is tangent to circle P at A, mAC = 60, mCD = 70,
and mDE = 80
Find mDAB, mCBA, and
mDGC.
The vertex of CBA is outside the circle, so
mCBA

17. Example 1 Given: AB is tangent to circle P at A, mAC = 60, mCD = 70,
and mDE = 80
Find mDAB, mCBA, and
mDGC.
The vertex of DGC is inside the circle, so
mDGC

18. Practice Problem Given: XY is tangent to circle T
at X, mUV = 90, mVX = 130,
and mUW = 20 V X T W Z Y 1 2
Find mWXY, m1, and
m2.

19. Homework
p.593 #10-26