1. 8-5 Angles in Circles

2. Central Angles
A central angle is an angle whose vertex is the CENTER of the circle
Central Angle
(of a circle)
Central Angle
(of a circle)
NOT A Central Angle
(of a circle)

3. CENTRAL ANGLES AND ARCS
The measure of a central angle is equal to the measure of the intercepted arc.

4. CENTRAL ANGLES AND ARCS
The measure of a central angle is equal to the measure of the intercepted arc.
Central Angle Y Z O
110
Intercepted Arc

5. EXAMPLE
Segment AD is a diameter. Find the values of x and y and z in the figure.
x = 25°
y = 100°
z = 55° A B O C D
55 x y
25 z

6. SUM OF CENTRAL ANGLES
The sum of the measures fo the central angles of a circle with no interior points in common is 360º.
360º

7. Find the measure of each arc.
2x-14 4x 2x 3x E B
3x+10
4x + 3x + 3x x + 2x – 14 = 360 …
x = 26
104, 78, 88, 52, 66 degrees A

8. Inscribed Angles
An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords. 3 1 2 4
Is NOT!
Is SO!
Is NOT!
Is SO!

9. INSCRIBED ANGLE THEOREM
Thrm 9-7. The measure of an inscribed angle is equal to ½ the measure of the intercepted arc.
The measure of an inscribed angle is equal to ½ the measure of the intercepted arc. x
1 2 x

10. INSCRIBED ANGLE THEOREM
Thrm 9-7. The measure of an inscribed angle is equal to ½ the measure of the intercepted arc.
The measure of an inscribed angle is equal to ½ the measure of the intercepted arc.

11. INSCRIBED ANGLE THEOREM
Thrm 9-7. The measure of an inscribed angle is equal to ½ the measure of the intercepted arc.
The measure of an inscribed angle is equal to ½ the measure of the intercepted arc.
Inscribed Angle Y
110
55 Z
Intercepted Arc

12. Find the value of x and y in the figure.
Thrm 9-7. The measure of an inscribed angle is equal to ½ the measure of the intercepted arc.
Find the value of x and y in the figure.
X = 20°
Y = 60° P
40 Q
50 y S x R T

13. Find the value of x and y in the figure.
Corollary 1. If two inscribed angles intercept the same arc, then the angles are congruent..
Find the value of x and y in the figure.
X = 50°
Y = 50° P Q y
50 S R x T

14. An angle formed by a chord and a tangent can be considered an inscribed angle.2x

15. An angle formed by a chord and a tangent can be considered an inscribed angle.P Q S R
mPRQ = ½ mPR

16. What is mPRQ ? P Q
60 S R

17. An angle inscribed in a semicircle is a right angle.P
180 R

18. An angle inscribed in a semicircle is a right angle.P
180
90 S R

19. Interior Angles
Angles that are formed by two intersecting chords. (Vertex IN the circle) A D B C

20. Interior Angle Theorem
The measure of the angle formed by the two chords is equal to ½ the sum of the measures of the intercepted arcs.

21. Interior Angle Theorem
The measure of the angle formed by the two chords is equal to ½ the sum of the measures of the intercepted arcs. 1 A B C D

22. Interior Angle Theorem
91 A C x° y° B D
85

23. Exterior Angles
An angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside the circle. (vertex OUT of the circle.)

24. Exterior Angles
An angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside the circle. k j j 1 k 1 k j 1

25. Exterior Angle Theorem
The measure of the angle formed is equal to ½ the difference of the intercepted arcs. 1 j k 3

26. Find
<C = ½(265-95)
<C = ½(170)
m<C = 85°

27. PUTTING IT TOGETHER! AF is a diameter. mAG=100 mCE=30 mEF=25
Find the measure of all numbered angles. Q G F D E C 1 2 3 4 5 6 A

28. Inscribed Quadrilaterals
If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary. P Q
mPSR + mPQR = 180  S R