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Angles and Arcs. Recognize major arcs, minor arcs, semicircles, and central angles and their measures. Find arc length. An artist’s rendering of Larry Niven’s Ringworld. Two artist’s conceptions of views from the surface of Ringworld. ANGLES AND ARCS.

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Angles and Arcs

  • Recognize major arcs, minor arcs, semicircles, and central

  • angles and their measures.

  • Find arc length.

An artist’s rendering of Larry Niven’s Ringworld.



ANGLES AND ARCS

A central angle has the center of a circle as its vertex, and its sides contain two radii of the circle.

The sum of the measures of the angles around the center of a circle is 360°.


Key Concept Sum of Central Angles

The sum of the measures of the central angles of a circle with no interior points in common is 360°.

1

3

2


Example 1 Measure of Central Angles

a) Find mAOB

B

3x°

C

25x°

2x°

D

A

O

E

b) Find mAOE


Key Concept Arcs of a Circle

Minor Arc

Major Arc

Semicircle

A

D

K

E

J

60°

110°

C

B

G

N

L

F

M

Usually named using the letters of the two endpoints.

Named by the letters of the two endpoints and another point on the arc.

Named by the letters of the two endpoints and another point on the arc.

AC

DFE

JML and JKL


Key Concept Arcs of a Circle

Minor Arc

Major Arc

Semicircle

JKL

A

D

DFE

AC

K

E

J

60°

110°

C

B

G

N

L

F

M

JML

Usually named using the letters of the two endpoints.

Named by the letters of the two endpoints and another point on the arc.

Named by the letters of the two endpoints and another point on the arc.

AC

DFE

JML and JKL


Theorem

In the same or in congruent circles, two arcs are congruent if and only if their corresponding central angles are congruent.


Postulate Arc Addition Postulate

The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

P

S

Q

R

In circle S, mPQ + mQR = mPQR


Example 2 Measures of Arcs

C

a) Find mBE

D

50°

B

A

F

b) Find mCBE

E

c) Find mACE


Example 3 Circle Graphs

BICYCLES

This graph shows the percent of each type of bicycle sold in the United States in 2001.

Identify any arcs that are congruent.


ARC LENGTH

Another way to measure an arc is by its length.

An arc is part of the circle, so the length of an arc is part of the circumference.


Example 4 Arc Length

In circle P, PR = 15 and mQPR = 120°. Find the length of arc QR.

Q

Solution:

r = 15, so C = 215 or 30, and

arc QR = mQPR or 120°.

Write a proportion to compare each part to its whole.

120°

P

15

R

l 31.42 units


Key ConceptArc Length

The proportion in the last example can be adapted to find an arc length in any circle.

degree measure of an arc 

degree measure of whole circle 

  • arc length

  • circumference

This can also be expressed as


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