CENTRAL AND INSCRIBED ANGLES - Spring-Ford …

[Pages:2]CENTRAL AND INSCRIBED ANGLES

A central angle is an angle whose vertex is the center of a circle and whose sides intersect the circle. The degree measure of a central angle is equal to the degree measure of its intercepted arc. For the circle at right with center C, ACB is a central angle.

An INSCRIBED ANGLE is an angle with its vertex on the circle and whose sides intersect the circle. The arc formed by the intersection of the two sides of the angle and the circle is called an INTERCEPTED ARC. ADB is an inscribed angle, A!B is an intercepted arc.

The INSCRIBED ANGLE THEOREM says that the measure of any inscribed angle is half the measure of its intercepted arc. Likewise, any intercepted arc is twice the measure of any inscribed angle whose sides pass through the endpoints of the arc.

mADB = 1 A!B and A!B = 2mADB

2

#18

A

C

B

A

?

?B ?

D

Example 1

In P, m !CPQ = 70? . Find mC!Q and mC! RQ .

C

Example 2

In the circle shown below, the vertex of the angles is at the center. Find x.

B

2x

R

P

Q

mC!Q = m !CPQ = 70?

C x A

mC! RQ = 360? ! mC"Q = 360? = 70? = 290?

Example 3

Since AB is the diameter of the circle,

x + 2x = 180? 3x = 180? x = 60?

Solve for x.

100? Ax

B

Since mB! AC = 220?, mB!C = 360? ? 220? = 140?. The C m!BAC equals half the mB!C or 70?.

120?

GEOMETRY Connections

51

? 2007 CPM Educational Program. All rights reserved.

Find each measure in diameters.

1. Y!Z 2. W!X

3. !VPZ 4. V! WX

P if m!WPX = 28? , m!ZPY = 38? , and WZ and XV are

5. !XPY 6. X!Y 7. X! WY 8. W! ZX

W

V

P

X

Z Y

In each of the following figures, O is the center of the circle. Calculate the values of x and justify your answer.

9.

10.

11.

12.

O 136? x

O 146? x

O x 49?

O x 62?

13.

14.

O x

100?

150? O

x

100?

17.

18.

55? O?

x

?O 12?

91?

x

15.

O ?

18? x

19.

O ?

35? x

16.

O ?

x 27?

20.

x O?

29? 101?

Answers

1. 38? 6. 114? 11. 98? 16. 27?

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2. 28? 7. 246? 12. 124? 17. 55?

3. 28? 8. 332? 13. 50? 18. 77?

4. 180? 9. 68? 14. 55? 19. 35?

5. 114? 10. 73? 15. 18? 20. 50?

Extra Practice

? 2007 CPM Educational Program. All rights reserved.

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