1) The document discusses angles and arcs related to circles, including how to find measures of central angles, inscribed angles, and angles formed by intersecting lines or secants. 2) Key terms are defined, such as minor/major arcs, semicircles, and inscribed angles. Properties are outlined, such as intercepted arcs determining angle measures. 3) Examples demonstrate finding angle measures using properties like intercepted arcs being equal to twice the angle measure for inscribed angles. Formulas are given for determining angle measures in different circle scenarios.

1. Angles and Arcs
The student is able to (I can):
• Find the measure of an arc or angle created by a central
angle
• Find the measure of an inscribed angle or the arc created
by one
• Find the measures of angles formed by lines that intersect
circles or the arcs created by them
• Use angle measures to solve problems

2. More vocabulary:
centralcentralcentralcentral angleangleangleangle – an angle whose vertex is on the center of the
circle, and whose sides intersect the circle.
minorminorminorminor arcarcarcarc – an arc created by a central angle less than 180˚.
It can be named with 2 or 3 letters.
major arcmajor arcmajor arcmajor arc – an arc created by a central angle greater than
180˚. Named with 3 letters.
semicirclesemicirclesemicirclesemicircle – an arc created by a diameter. (= 180˚)

3. Example:
•
S
P
A
T
(or )SP PS
is a minor arc.
(or )SAP PAS
is a major
arc.
SPA is a semicircle.
•
•
•
∠PTA and ∠STP are
central angles.

4. Examples
Find the measure of each
1.
2.
3. m∠CAT
•
•
135˚
F
R
E
D
mRE
mEFR
C
A
T
260˚

5. Examples
Find the measure of each
1. = 135˚
2. = 360 – 135
= 225˚
3. m∠CAT
•
•
135˚
F
R
E
D
mRE
mEFR
C
A
T
260˚
= 360 – 260
= 100˚

6. If a radius or diameter is perpendicular to a chord, then it
bisects the chord and its arc.
ER GO⊥
G
E
O
R
A
GA AO≅
GR RO≅

7. Example
Find the length of .BU
•
B
L
U
E2
5
x

8. Example
Find the length of .BU
•
B
L
U
E
3333
2
5
2 2 2
3 5x+ =
x
x = 4
BU = 2(4) = 8

9. 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= ⋅ ∠

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

11. 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= ∠

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

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

14. 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°

15. 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∠ =
•

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

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

18. 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∠ = +

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

20. 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°

21. 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∠ = −

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

23. 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°

24. 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= −

25. 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)˚

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

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