Download Geometry Honors Section 9.3 Arcs and Inscribed Angles

Geometry Honors Section 9.3
Arcs and Inscribed Angles
Recall that a *central angle is an angle
whose vertex is at the center of the
circle and whose sides are radii.
What is the relationship between a
central angle and the arc that it cuts
off?
The measure of the central angle
equals the measure of its
intercepted arc.
An *inscribed angle is an angle
whose vertex lies on the circle and
whose sides are chords.
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.
Given the measure of 1, complete the table.
Remember that the radii of a circle are congruent.
0
0
0
20 40 40
0
0
0
30 60 60
0
0
0
x 2x 2x
What does the table show about the
relationship between m1 and mPK ?
1
m1 
mPK
2
Inscribed Angle Theorem
The measure of an angle inscribed
in a circle is equal to ½ its
intercepted arc.
35
0
35
0
90
0
700
Corollaries of the Inscribed Angle Theorem:
If two inscribed angles intercept the
same arc, then the angles are congruent.
If an inscribed angle intercepts a
semicircle, then the angle is a right angle.
0
65
0
65
1300
500
0
35
0
35
50
0
0
90
0
120
700
1100
A second type of angle that has its vertex on the circle is an angle formed by
a tangent and a chord intersecting at the point of tangency.
0
0
120 30
0
0
100 40
0
0
80
50
90
0
90
0
90 0
0
60
0
50
0
40
Theorem: If a tangent and a chord
intersect on a circle at the point of
tangency, then the measure of the
angle formed is equal to ½ the
measure of the intercepted arc.
75
0
194
0
1660
830