Download Day 36 – Inscribed Angles

Warm up
Case I:
Central Angle: Vertex is
A
AT the center
P

B
Central ANGLE = ARC
Case II:
Inscribed Angle:
Vertex is ON circle

ANGLE
ARC
ANGLE =
2
ARC

Intercepted Arc:
• An arc whose endpoints are the two
points of intersection of an angle
with the circle and all points that lie
within the angle.
Intercepted Arc
Inscribed Angle 
2
160°
80°
The arc is
twice as big as
the angle!!
Find the value of x and y
120

x
y


Examples
1. If mJK = 80 and JMK = 2x – 4, find x.
x = 22
2. If mMKS = 56, find m MS.
112 
J
K
Q
M
S
Find the measure of DOG and DIG
72˚
D
If two inscribed
angles intercept
the same arc,
then they are
congruent.
G
O
I
If all the vertices of a polygon
touch the edge of the circle, the
polygon is INSCRIBED and the
circle is CIRCUMSCRIBED.
Circumscribed Circle
• The circumscribed circle (or
circumcircle) of a polygon is a circle
which passes through all the vertices of
the polygon.
• The center of this circle is called the
circumcenter and its radius is called the
circumradius.
a quadrilateral inscribed in a
circle: opposite angles are
supplementary.
B
A
D
C
mA  mC  180
mB  mD  180
If a right triangle is inscribed in a
circle then the hypotenuse is the
diameter of the circle.
Example 3
In J, m3 = 5x and m 4 = 2x + 9.
Find the value of x.
Q
D
x=3
T
3
J
4
U
Example 4
In K, GH is a diameter and mGNH = 4x – 14.
Find the value of x.
4x – 14 = 90
x = 26
H
K
G
N
Bonus: What type of triangle is this? Why?
Example 5 Find y and z.
z
110
110 + y =180
y
y = 70
z + 85 = 180
z = 95
85