1. 8-5 Angles in Circles
Welcome everyone!

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.
Central Angle Y Z O
110
Intercepted Arc

4. Minor Arc An Arc is part of the circle.
A Minor Arc is an arc above the central angle if the central angle is less then 180°

5. Major Arc
A Major Arc is an arc above the central angle if the central angle is greater then 180°

6. Measure of an Arc
The measure of an Arc is the same as the central angle.

7. Measure of an Arc
The measure of an Arc is the same as the central angle.

8. 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

9. Semicircle
If the central angle equals 180°, then the arc is a semicircle.

10. 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!

11. Definition
Chord – a segment whose endpoints are points on the circle.

12. 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

13. 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.

14. 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

15. 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

16. 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

17. Definition
Tangent – a line in the plane of a circle that intersects the circle in exactly one point.

18. Definition
Secant – a line that intersects a circle in two points.

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

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

21. What is mPRQ ? P Q
60 S R

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

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

24. 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

25. 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, 38 degrees A