Download PP Section 10.5

Geometry Section 10.5
Angle Relationships in Circles
What you will learn:
1. Find angle and arc measures
2.Use circumscribed angles
The location of an angle’s vertex determines
the relationship between the angle and its
intercepted arcs.
I. Vertex at the center

m  m
II. Vertex on the circle
1 
m  m
2
Theorem 10.14: Tangent and Intersected Chord
Theorem
If a chord and a tangent intersect at a point on the
circle, then then the measure of each angle
formed is one half the measure of the intercepted
arc.
C
B
1
m1  mABC
2
1
m2  mAC
2
12
A
60
160
30 80
110
110
50
140
30 80
25 70
110
95
110
95
40
120
30 80 110
25 70 95
20 60 80
110
95
80
Theorem10.15: Angles Inside the Circle
Theorem
The measure of an angle formed by two
chords that intersect inside a circle is equal to
one-half the sum of the measures of the arcs
intercepted by the angle and its vertical
angle.
mAVC  1 80  40
2
mAVC  60
mAVB  120
88  1 120  mBD 
2
176  120  mBD 
mBD  56
IV. Vertex outside the circle
Theorem 10.16: Angles Outside the Circle
Theorem
The measure of an angle formed by two lines
that intersect outside a circle is equal to onehalf the difference of measures of the
intercepted arcs.
IV. Vertex outside the circle
A
C
A
C
1
D
B
1
m1  mAB  CD 
2
1
C
A
1
B
1
m1  mCB  AB 
2
B
1
m1  mACB  AB 
2
113
67
mAVC  1 80  20
2
mBVC  1 113  67 
2
mAVC  1 60   30
2
mBVC  1
2
46  23
HW: pp 566 & 567 / 3 – 14, 17 – 22, 25