Chapter 10 – Circles
Section 10.3 – Inscribed Angles
Unit Goal
Use inscribed angles to solve problems.
D
A
C
B
Basic Definitions
INSCRIBED ANGLE – an angle whose vertex is
on the circle
INTERCEPTED ARC – the arc whose endpoints
D
are are on the inscribed angle
DCB is an inscribed angle.
DB is the intercepted arc.
A
C
B
What Is the Measure of
an Inscribed Circle?
What is the measure of BEC ?
What is the measure of BDC ?
B
A
D
C
E
Theorem 10.8
Measure of an Inscribed Angle
The measure of an inscribed angle is ½ of its
intercepted arc.
B
1
m BDC  mBC
2
A
D
C
Example
Find the measure of the angle or arc:
C
D
D
A
A
B
I
C
20º
m BDC  140
mBIC 
B
m CDB 
Example
Find the measure of the angle or arc:
B
B
A
A
E
D
D
C
m BDC 
mBEC 
50º
C
mBC 
Example
B
60º
C
D
J
K
m BDC 
m BJC 
m BKC 
Theorem 10.9
If two inscribed angles of a circle intercept the
same arc, then the angles are congruent.
B
60º
C
D
J
K
Properties of Inscribed Polygons
If all the vertices of a polygon lie on a circle, the
polygon is INSCRIBED in the circles and the
circle is CIRCUMSCRIBED about the polygon
Theorems About Inscribed Polygons
Theorem 10.10
If a right triangle is inscribed
in a circle, then the
hypotenuse is a diameter of
the circle. Conversely, if one
side of an inscribed triangle
is a diameter of the circle,
then the triangle is a right
triangle and the angle
opposite the diameter is the
right angle
<B is a right angle iff
segment AC is a diameter of
the circle
A
B
P
C
Theorem 10.11
A quadrilateral can be
inscribed in a circle iff its
opposite angles are
supplementary
D, E, F, and G lie on
some circle C iff
m<D + m<F = 180° AND
m<E + m<G = 180°
F
E
C
D
G
Example
A
2y
In the diagram, ABCD is inscribed
in circle P. Find the measure of
each angle.
ABCD is inscribed in a circle, so
opposite angles are supplementary
3x + 3y = 180 and 5x+ 2y = 180
3x + 3y = 180 (solve for x)
- 3y -3y
3x
= -3y + 180
3
3
x = -y + 60
Substitute
3y
D
P
3x
5x
C
Substitute this into the second
equation 5x + 2y = 180
5 (-y + 60) + 2y = 180
-5y + 300 + 2y = 180
-3y = -120
y = 40
x = -y + 60
x = -40 + 60 = 20
B
Example (cont.)
x = 20, y = 40
m<A = 2y, m<B = 3x, m<C = 5x, m<D = 3y
m<A = 80°
m<B = 60°
m<C = 100°
m<D = 120°
HW Assignment
p. 616-617 (4 – 28 even)