1. Section 9-5: Inscribed Angles & Corollaries

2. ÐABC is an inscribed angle of Circle O.
Definition: an Inscribed Angle is an angle with its vertex on the circle. A O C B
ÐABC is an inscribed angle of Circle O.

3. This inscribed angle intercepts an arc of Circle O. The intercepted
110°
The measure of the intercepted arc of an inscribed angle is equal to twice the measure of the inscribed angle. O C
55° B
This inscribed angle intercepts an arc of Circle O. The intercepted
arc is AC.

4. ÐABC and ÐADC both intercept AC.
Corollary 1: Inscribed angles that intercept the same arc are congruent. A D C B
100°
ÐABC and ÐADC both intercept AC.
50°
50°
mÐABC = 50
mÐADC = 50

5. Corollary 2: An angle inscribed inside of a semicircle is a right angle.
110°
70°
mÐBAC = 90
35°
55°
Here’s why…

6. Corollary 3: If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. A D C B
mÐBAC = 76
76°
92°
mÐACD = 92
88°
104°

7. Section 9-6: Other Angles

8. Theorem 9-10: The measure of an angle formed by two secants, two tangents, or a secant & a tangent is equal to half the difference of the intercepted arcs.
mÐABC = ½(mAC – mDE). A D C B E

9. Case 1:
Two Secants A D C B E
84°
20°

10. Case 2: Two Tangents
mÐADB = ½(226 – 134) A D B
mÐADB = 46
226°
134°

11. Case 3:
A secant & a tangent A D C B
30°
128°

12. mAG = 100 mCE = 30 mEF = 25 A D C B E F O G 3 7 8 2 1 4 5 6
Given: AB is tangent to Circle O; AF is a diameter.
Find all numbered angles. A D C B E F O G 3 7 8 2 1 4 5 6
mAG = 100
mCE = 30
mEF = 25

13. m1 = ___
m2 = ___
m3 = ___
m4 = ___
m5 = ___
m6 = ___
m7 = ___
m8 = ___