Download Geometry Section 10.4 Inscribed Angles and Polygons

Geometry Section 10.4
Inscribed Angles and Polygons
What you will learn:
1. Use inscribed angles
2.Use inscribed polygons
Recall that a *central angle is an angle whose
vertex is at the center of the circle.
What is the relationship between a central angle
and the arc it cuts off?
central angle  intercepted arc
An *inscribed angle is an angle whose vertex
lies on the circle and whose sides contain
chords of the circle.
A
E
T
By doing the following activity, you will be
able to determine the relationship between
the measure of an inscribed angle and the
measure of its intercepted arc.
20
30
x
40
60
2 x
40
60
2 x
m1  1 mPK
2
Theorem 10.10 Measure of an Inscribed Angle
Theorem
The measure of an inscribed angle is one-half the
measure of its intercepted arc.
Example: Find the value of x in each figure. Q is the
center of each circle.
89
59
78
1
x  89  44.5
2
1
x  59  29.5
2
x  180  78  102
130
65
65
35
35
50
90
120
50
70
110
128
236
232
124
116
62
116
118
74
This work suggests the following theorem.
Theorem 10.13: Inscribed Quadrilateral
Theorem
A quadrilateral can be inscribed in a circle
(i.e. its vertices lie on the circle) if and only if
its opposite angles are supplementary.
Example: Find the value of each variable in the
figure at the right.
3a  111  180
3a  69
a  23
b  65  180
b  115
HW: pp 558 & 559 / 3 – 16, 25 - 30