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6.5 Apply Other Angle Relationships
Theorem 6.13If 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.
_____ 2
11m AB
A
C
B
12_____ 2
12 m BCA
6.5 Apply Other Angle Relationships
Example 1 Find angle and arc measuresFind angle and arc measures
Line m is tangent to the circle. Find the indicated measure.
1 a. m
m
A
B
1
o132
Solution
____1321 a. o
_____m
2
1 o66
EFD 2. mm
F
D o110
E
____110__EFD b. o m 2 o220
6.5 Apply Other Angle RelationshipsCheckpoint. Find the indicated measure. Checkpoint. Find the indicated measure.
ACB 1. m
ABo120
C
ACBm o1202 o240
6.5 Apply Other Angle Relationships
Theorem 6.14If two chords intersect inside a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
______ 2
11 mmm DC D
C
A
B
12
AB
______ 2
12 mmm AD BC
6.5 Apply Other Angle Relationships
Example 2 Find an angle measure inside a circleFind an angle measure inside a circle
Find the value of x.
Solution
J
H
F
G
ox
o112
o140
The chords FH and GJ intersect inside the circle.
______ 2
1o mmx FJ HG
________ 2
1o x o112 o140
____x 126
6.5 Apply Other Angle RelationshipsCheckpoint. Find the value of Checkpoint. Find the value of xx..
2.
J
H
F
G
o144
o52
ox
2
1 128o ox o1442 2
ooo 144256 x
oo 52180 o128
x 112
6.5 Apply Other Angle Relationships
Theorem 6.15If 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.
1
BA
C
ACBC 2
11 mmm
Q
P
R
2
PRPQR 2
12 mmm
3
X
Y
W
Z
Zmmm WXY 2
13
6.5 Apply Other Angle Relationships
Example 3 Find an angle measure outside a circleFind an angle measure outside a circle
Find the value of x.
SolutionThe tangent GF and the secant GJ intersect outside the circle.
______ 2
1FGH mmm FJ FH
________ 2
1o x o170 o72
____x 49
oxo72
o170
J
F
H
G
6.5 Apply Other Angle RelationshipsCheckpoint. Find the value of Checkpoint. Find the value of yy..
2
1 26o oy o302 2
ooo 3052 y
y 82
3.
J
K
H
G
o30
o26
oy
F
6.5 Apply Other Angle Relationships
Pg. 225, 6.5 #1-23
