Find the measures of angles formed by lines that intersect circles. Use angle measures to solve problems.

1. 6.14.3 Angle Relationships
The student is able to (I can):
• Find the measures of angles formed by lines that
intersect circles
• Use angle measures to solve problems

2. If a tangent and a secant (or a chord)
intersect at the point of tangency, then
the measure of the angle formed is half the
measure of its intercepted arc.
F
L
•
Y
LF is a secant.
LY is a tangent.
∠ =
1
m FLY mFL
2

3. Example Find each measure:
1. m∠EFH
2.
180 — 122 = 58º
mGF
∠ = = °
1
m EFH (130) 65
2
58º
= = °mGF 2(58) 116

4. If two secants or chords intersect in the
interior of a circle, then the measure of
each angle formed is half the sum of the
intercepted arcs.
1111
G
R
A
D
( )∠ = +
1
m 1 mDG mRA
2

5. Example Find each measure.
1. m∠1
2. m∠2
m∠2 = 180 — m∠1
= 180 — 80 = 100º
99º
61º
1
2
( )∠ = +
1
m 1 99 61
2
= 80º

6. If secants or tangents intersect outside a
circle, the measure of the angle formed is
half the difference between the intercepted
arcs.
M O N
E
Y
1
( )∠ = −
1
m 1 mNY mOE
2

7. Example Find each measure
1. m∠K
2. x
186º
62º
K
26º
94º
∠ = −
1
m K (186 62)
2
= 62º
= −
1
26 (94 x)
2 xº
52 = 94 — x
x = 42º

8. Like the other angles outside a circle, if two
tangents intersect outside a circle, the
measure of the angle formed is half the
difference between the intercepted arcs.
Unlike the other angles, however, because
the two arcs addaddaddadd to 360˚, we can use
algebra to simplify things a little.
y˚
x˚
(360-x)˚
− − −
= =
= −
360 x x 360 2x
y
2 2
y 180 x
or
= −x 180 y

9. Example: Solve for x.
x˚ 64˚
= − = °x 180 64 116

10. If we are trying to find the outer arc, flip
around the x˚ and (360-x)˚ and re-write the
equation:
Example: Solve for x.
( )− − − +
= =
−
=
= −
x 360 x x 360 x
y
2 2
2x 360
2
y x 180 or = +x 180 y
y˚
x˚
(360-x)˚
x˚51˚
= + = °x 180 51 231