Lesson 9-4
Objective: To recognize and
find measures of inscribed
angles, and to apply properties of
inscribed figures.
Inscribed Angle
Inscribed Angle: An angle whose vertex lies on a circle and whose
sides are chords of the circle (or one side tangent to the circle).
ABC is an inscribed angle.
B
O
Examples:
1
C
A
2
3
D
4
Intercepted Arc
Intercepted Arc: An angle intercepts an arc if and only if :
1. The endpoints of the arc lie on the angle.
2. All points of the arc, except the endpoints, are in the
interior of the angle.
3. Each side of the angle contains an endpoint of the arc.
C
B
ADC is the int ercepted arc of ABC.
Think of Pac Man, his mouth
opening is the intercepted arc.
O
A
D
Inscribed Angle Theorem
The measure of an inscribed angle is equal to ½
the measure of the intercepted arc.
Y
Inscribed Angle
55
Z
Intercepted Arc
Example
Name an inscribed angle?
Name an arc intercepted
by angle BAC.
If m BPC = 42,
find the m BAC. What is the
measure of arc BC?
Another Example

Find X
And Another Example
If the measure of arc ST=68
R
Find the measure of angle
1
1 and angle 2.

S
Q
Note: Q is not the center
of the circle.
T
2
P
One Last Example…
Arc KL is 40
Find all
the angle
measures.
K
6 1
2 3
L
Arc LP = 100
M
4
P
5
N
Arc NP is 60
Theorem
If two inscribed angles of a circle intercept
congruent arcs (or the same arc), then the
angles are
congruent.
Theorem: An angle inscribed in a
semicircle is a right angle.
P
S
180
90
R
Example Using Theorem
If the measure
of angle HTC = 52°,
find the
measure of
angle CEH.
Why?
Which angle do you
think is 90 °?

Example Using Theorem
If m 4 = 7x +3 and
m 5 = 7x +3 and
m 1 = 5x. Find x
4
and all the
1
measures of the other
angles.

6
5
2
3
Theorem: Inscribed
Quadrilaterals
If a quadrilateral is inscribed in a circle, then the opposite
angles are supplementary.
mB + mD = 180 
mA+ mC = 180 
Last One….
A
B
D
C
Make up an example
where one angle is
given and the
opposite angle is
unknown.
Remember:
mD + mB = 180 
mA + mC = 180 