Download 10.5 Apply Other Angle Relationships in Circles

10.5 Apply Other Angle
Relationships in Circles
Hubarth
Geometry
Theorem 10.11
If a tangent and a chord intersect at a point on a circle, then the
measure of each angle formed is one half the measure of its
intercepted arc.
C
.
B
2 1
A
m1 
1
mAB
2
m2 
1
mBCA
2
Ex 1 Find Angle and Arc Measures
Line m is tangent to the circle. Find the measure of the red angle or arc.
a.
m
1 =
1
2
(130o)
= 65o
b.
m KJL
=
2(125o)
= 250o
Theorem 10.12 Angles Inside the Circle Theorem
D
If two chords intersect inside a circle, then the measure
of each angle is one half the sum of the measure of the
arcs intercepted by the angle and its vertical angle.
A
1
2
B
C
m1 
1
(mDC  mAB)
2
m2 
1
(mAB  mBC )
2
Theorem 10.13 Angles Outside the Circle Theorem
If a tangent and a secant, two tangents or two secants intersect outside a circle, then the
measure of the angle formed is one half the difference of the measures of the intercepted
arcs.
P
X
B
A
Q
W
3
1
2
Z
Y
R
C
1
1
1
m

3

(mXY  mWZ )
m2  (mPQR  mPR)
m1  (mBC  mAC )
2
2
2
.
Ex 2 Find an Angle Measure Inside a Circle
Find the value of x.
xo =
x
o
=
1
(mJM + mLK)
2
1
o
o
(130
+
156
)
2
xo = 143
Ex 3 Find and Angle Measure Outside a Circle
Find the value of x.
The tangent CD and the secant CB intersect outside the
circle.
m
1
BCD = 2 (mAD – mBD)
xo =
x
1
o
o
(178
–
76
)
2
= 51
Practice
Find the indicated measure.
m
1=
1
2
(210o) = 105o
m RST = 2(98o) = 196o
m XY =
o
2(80o) = 160
Find the value of the variable.
5.
o
y = 61
6.
a
= 104o
xo
253.7o