angles in circles

1. Angles in Circles

2. Central Angles
Central
Angle
(of a circle)
Central
Angle
(of a circle)
NOT A
Central
Angle
(of a circle)
• A central angle is an angle whose vertex is
the CENTER of the circle

3. CENTRAL ANGLES AND ARCS
The measure of a central angle is equal to
the measure of the intercepted arc.

4. CENTRAL ANGLES AND ARCS
The measure of a central angle is equal to
the measure of the intercepted arc.
Y
Z
O 110
1
1
0

Intercepted Arc
Central
Angle

5. EXAMPLE
• Segment AD is a diameter. Find the
values of x and y and z in the figure.
x = 25°
y = 100°
z = 55°
A
B
O
C
D
55
x y
25
z

6. SUM OF CENTRAL ANGLES
The sum of the measures fo the central angles of a circle with
no interior points in common is 360º.
360º

7. Find the measure of each arc.
A
E B
C
D
2x
2
x
-
1
4
4
x
3x
3
x
+
1
0
4x + 3x + 3x + 10+ 2x + 2x – 14 = 360
…
x = 26
104, 78, 88, 52, 66 degrees

8. An inscribed angle is an angle
whose vertex is on a circle and
whose sides contain chords.
Inscribed Angles
1 4
2
3
Is
NOT!
Is
NOT!
Is SO! Is SO!

9. Thrm 9-7. The measure of an inscribed angle is
equal to ½ the measure of the intercepted arc.
INSCRIBED ANGLE THEOREM
The measure of an inscribed angle is equal
to ½ the measure of the intercepted arc.

10. Thrm 9-7. The measure of an inscribed angle is
equal to ½ the measure of the intercepted arc.
INSCRIBED ANGLE THEOREM
The measure of an inscribed angle is equal
to ½ the measure of the intercepted arc.
1
2


11. Thrm 9-7. The measure of an inscribed angle is
equal to ½ the measure of the intercepted arc.
INSCRIBED ANGLE THEOREM
The measure of an inscribed angle is equal
to ½ the measure of the intercepted arc.
Y
Z
55
1
1
0

Inscribed Angle
Intercepted Arc

12. Thrm 9-7. The measure of an inscribed angle is
equal to ½ the measure of the intercepted arc.
x
y
Q
R
P
S
T
50
40
Find the value of x and y
in the figure.
• X = 20°
• Y = 60°

13. Corollary 1. If two inscribed angles intercept the
same arc, then the angles are congruent..
x R
Q
S
T
50
P
y
Find the value of x and y
in the figure.
• X = 50°
• Y = 50°

14. An angle formed by a chord and a tangent
can be considered an inscribed angle.

15. An angle formed by a chord and a tangent
can be considered an inscribed angle.
R
S
P
Q
mPRQ = ½ mPR

16. What is mPRQ ?
R
S
P
Q
60

17. An angle inscribed in a
semicircle is a right angle.
R
P 180

18. An angle inscribed in a
semicircle is a right angle.
R
P 180
S
90

19. • Angles that are formed by two
intersecting chords. (Vertex IN the
circle)
Interior Angles
A
B
C
D

20. Interior Angle Theorem
The measure of the angle formed by the
two chords is equal to ½ the sum of the
measures of the intercepted arcs.

21. Interior Angle Theorem
The measure of the angle formed by the
two chords is equal to ½ the sum of the
measures of the intercepted arcs.
1
A
B
C
D 1
m 1 (mAC mBD)
2
  

22. A
B
C
D
x°
91
85
Interior Angle Theorem
91 5
(
2
8
1
)
x  
88

x
y°
88
180
 
y
92

y

23. • An angle formed by two secants, two
tangents, or a secant and a tangent
drawn from a point outside the circle.
(vertex OUT of the circle.)
Exterior Angles

24. • An angle formed by two secants, two
tangents, or a secant and a tangent
drawn from a point outside the circle.
Exterior Angles
1
j
k 1
j
k
1
j
k

25. Exterior Angle Theorem
• The measure of the angle formed is equal
to ½ the difference of the intercepted
arcs.
1
j
k 3
j
k
1
j
k
1
m 1 (k j)
2
  

26. Find m ACB

• <C = ½(265-95)
• <C = ½(170)
• m<C = 85°
265
95
C
B
A

27. PUTTING IT TOGETHER!
• AF is a diameter.
• mAG=100
• mCE=30
• mEF=25
• Find the measure
of all numbered
angles.
Q
G
F
D
E
C
1
2
3
4
5
6
A

28. R
S
P
Q
Inscribed Quadrilaterals
• If a quadrilateral is inscribed in a circle,
then the opposite angles are supplementary.
mPSR + mPQR = 180 