1) When a tangent and secant or chord intersect at the point of tangency, the measure of the angle formed is half the measure of the intercepted arc. 2) When two secants or chords intersect in the interior of a circle, the measure of each angle formed is half the sum of the intercepted arcs. 3) When secants or tangents intersect outside a circle, the measure of the angle formed is half the difference between the intercepted arcs. Algebraic formulas are provided to calculate angle measures using the intercepted arcs.

1. 6.14.4 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