10.5 inscribed angles. an inscribed angle is an angle whose vertex is on a circle and whose sides...
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10.5 Inscribed Angles
An inscribed angle is an angle whose vertex is on a circle and whose sides are determined by two chords.
Intercepted arc: the arc that lies in the interior of an inscribed angle and has endpoints on the angle.
inscribed angle
Intercepted arc
A tangent-chord angle is an angle whose vertex is on a circle and whose sides are determined by a tangent and a chord that intersect at the tangents’ point of contact.
Theorem 86: The measure of an inscribed angle or a tangent-chord angle (vertex on a circle) is one-half the measure of its intercepted arc.
C
A
D
B
1
2m ADB mAB
A chord-chord angle is an angle formed by two chords that intersect inside a circle but not at the center.
Theorem 87- The measure of a chord-chord angle is one-half the sum of the measures of the arcs intercepted by the chord-chord angle and its vertical angle.
O
BA
D
P
C
1
2m CPD m CD AB
A secant-tangent angle is an angle whose vertex is outside a circle and whose sides are determined by a secant and a tangent.
A tangent-tangent angle is an angle whose vertex is outside a circle and whose sides are determined by two tangents.
A secant-secant angle is an angle whose vertex is outside a circle and whose sides are determined by two secants.
Theorem 88- The measure of a secant-tangent angle, a tangent-tangent angle, or a secant-secant angle (vertex outside the circle) is one-half the difference of the measures of the intercepted arcs.
There are three types of angles having a vertex outside a circle and both sides intersecting the circle.
B
C
A
1
1 is a secant-tangent angle
11
2m mBC mAC
P
Q
R
2
2 is a tangent-tangent angle
12
2m mPQR mPR
3
Y
Z
W
X
13
2m mXY mWZ
3 is a secant-secant angle
Example 1: Find the measure of the arc or angle.
C
B
D
A
mACB
70
C
B
A
mAC
196
CB
A
m ABC
Example 2: It is given that mB = 44. What is m C?
D
CB
A
Example 3: Line m is tangent to the circle. Find mRST
102
R
m
ST
Example 4: BC is tangent to the circle. Find CBD
4x+50 3x
D
A
BC
Example 5: Find the value of x.
Example 6: Find the value of x.
120
x
40
U
T
R
S
80x
S
T
R
P