The document outlines various angle relationships related to circles, including inscribed angles and their measures. It provides corollaries for angles formed by secants, chords, and tangents, along with examples and formulas for finding angle measures based on intercepted arcs. Key concepts include how to calculate measures of angles formed at tangents and secants, both inside and outside circles.

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 intersect
circles
• Use angle measures to solve problems

2. inscribedinscribedinscribedinscribed angleangleangleangle – 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.
•
H
A
I
R
1
2
m AHR AR∠ =
2AR m AHR= ⋅ ∠

3. Examples
Find each measure:
1. m∠MAP
2.
M
A
P
110°
J
Y
O
24°
mJY

4. Examples
Find each measure:
1. m∠MAP
2.
= 2(24)
= 48°
M
A
P
110°
J
Y
O
24°
mJY
( )1
m m
2
MAP MP∠ =
1
(110) 55
2
= = °
m 2(m )JY JOY= ∠

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

6. Corollary: An inscribed angle intercepts a semicircle if and
only if it is a right angle.
•

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

8. If a tangent and a secant (or 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
is a secant.LF
is a tangent.LY
1
m m
2
FLY FL∠ =
•

9. Examples
Find each measure:
1. m∠EFH
2. mGF
58°

10. Examples
Find each measure:
1. m∠EFH
2.
180 – 122 = 58°
mGF
1
m (130) 65
2
EFH∠ = = °
58°
m 2(58) 116GF = = °

11. 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.
1
G
R
A
D
( )1
m 1 m m
2
DG RA∠ = +

12. Examples
Find each measure.
1. m∠1
2. m∠2
99°
61°
1
2

13. Examples
Find each measure.
1. m∠1
2. m∠2
m∠2 = 180 – m∠1
= 180 – 80 = 100°
99°
61°
1
2
( )
1
m 1 99 61
2
∠ = +
= 80°

14. 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 m m
2
NY OE∠ = −

15. Examples
Find each measure
1. m∠K
2. x
186°
62°
K
26°
94°
x°

16. Examples
Find each measure
1. m∠K
2. x
186°
62°
K
26°
94°
1
m (186 62)
2
K∠ = −
= 62°
1
26 (94 )
2
x= −
x°
52 = 94 – x
x = 42°

17. 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)˚
360 360 2
2 2
180
x x x
y
y x
− − −
= =
= −
or
180x y= −

18. If we are trying to find the outer arc, flip around the x˚ and
(360-x)˚ and re-write the equation:
( )360 360
2 2
2 360
2
180
x x x x
y
x
y x
− − − +
= =
−
=
= − or 180x y= +
y˚
x˚
(360-x)˚

19. Examples: Solve for x.
1.
2.
x˚ 64˚
x˚51˚

20. Examples: Solve for x.
1.
2.
x˚ 64˚
180 64 116x = − = °
x˚51˚
180 51 231x = + = °