Find the measures of inscribed angles Find the measures of angles formed by tangent and secant lines

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 = + = °