Angles in

1
Angles in a Circle
Chapter 7 The Circle
7.3B
7.3.1
MATHPOWERTM 11, WESTERN EDITION
2
Angles in a Circle Theorems
1. The measure of the central angle is equal
to twice the measure of the inscribed
angle subtended by the same arc. 2.
Inscribed angles subtended by the same arc,
or by equal arcs, are congruent. 3. The
angle inscribed in a semicircle is a right angle.
C
O
Both angles are subtended by the arc AB.
B
A
7.3.2
3
Measure of the Arcs
The arc AB will have the same measure as that of
the central angle AOB.
O
550
400
700
A
B
700
400
550
C
The arc AB will be twice the measure of
the inscribed angle ACB.
300
400
B
A
7.3.3
600
800
4
Angles in a Circle - Theorem 1
The measure of the central angle is equal to
twice the measure of the inscribed angle
subtended by the same arc.
B
Inscribed Angle
Central Angle
Radii
AO BO
O
ITT
Exterior Angle Theorem
A
Substitution
C
Simplify
Prove that
7.3.4
5
Angles in a Circle - Theorem 2
Inscribed angles subtended by the same arc, or
by equal arcs, are congruent.
C
B
O
D
Angles in a Circle
A
Prove that
Angles in a Circle
Transitive
7.B.5
6
Angles in a Circle - Theorem 3
The angle inscribed in a semicircle is a right
angle.
C
B
Angles in a circle
O
Straight Angle
A
Substitution
Prove
7.3.6
7
Using the Angles in a Circle Theorems
Find the measure of the indicated angles.
1580
Note All of the angles are subtended by the
arc EF. The measure of arc EF 860.
D
430
B
A
430
O
430
580
580
860
Given more information.
790
E
F
790
860
7.3.7
8
Using the Angles in a Circle Theorems
A
B
C
900
7.3.8
9
Using the Angles in a Circle Theorems
900
B
200
620
640
C
590
O
310
400
A
390
580
D
E
620
7.3.9
10
Assignment
Suggested Questions
Pages 412 and 413 1-24, 29
7.3.10