- An inscribed angle is half the measure of its intercepted arc. If inscribed angles intercept the same arc, they are congruent. An angle is a right angle if it intercepts a semicircle. - For angles formed by intersecting secants or tangents at the point of tangency, the angle measure is half the intercepted arc. - When two secants or chords intersect in the interior of a circle, the angle measure is half the sum of the intercepted arcs. When they intersect outside the circle, the angle measure is half the difference of the intercepted arcs. - Opposite angles of an inscribed quadrilateral are supplementary.

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

2. inscribed angle An angle whose vertex is on the circle and
whose sides contain chords of the circle.
The measure of an inscribed angle is ½ the
measure of its intercepted arc.
A
∠ =
1
m AHR AR
•
H
I
R
∠ =
1
m AHR AR
2
= ⋅ ∠AR 2 m AHR

3. Example Find each measure:
1. m∠MAP
M
A
P
110º
( )∠ =
1
m MAP mMP
2
= = °
1
(110) 55
2
2.
= 2(24)
= 48º
J
Y
O
24º24º24º24º
mJY
= ∠mJY 2(m JOY)

4. If inscribed angles intercept the same arc,
then the angles are congruent.
R
E
∠RED ≅ ∠RAD
A D

5. An inscribed angle intercepts a semicircle if
and only if it is a right angle.
•

6. If a quadrilateral is inscribed in a circle, its
opposite angles are supplementary.
F
R
E
FRED is inscribed
in the circle.
D
m∠F + m∠E = 180º
m∠R + m∠D = 180º

7. 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
LF is a secant.
LY is a tangent.
L
•
Y
∠ =
1
m FLY mFL
2

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

9. 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
( )∠ = +
1
m 1 mDG mRA
2
A
D

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

11. 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
1
E
Y
( )∠ = −
1
m 1 mNY mOE
2

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