Download 10.3 Inscribed Angles

12.3 Inscribed Angles
Using Inscribed Angles
• An inscribed angle is
an angle whose
vertex is on a circle
and whose sides
contain chords of
the circle. The arcinscribed angle
that lies in the
interior of an
inscribed angle and
has endpoints on
the angle is called
the intercepted arc
of the angle.
intercepted arc
Theorem 12.9: Measure of an
Inscribed Angle
A
• If an angle is
inscribed in a
circle, then its
measure is one
half the measure
of its intercepted
arc.
mADB = ½m AB

C
D
B
Ex. 1: Finding Measures of Arcs
and Inscribed Angles
• Find the measure
of the blue arc or
angle.
S
R

m QTS = 2mQRS =
T
2(90°) = 180°
Q
Ex. 1: Finding Measures of Arcs
and Inscribed Angles
• Find the measure
of the blue arc or
angle.

m ZWX = 2mZYX =
2(115°) = 230°
W
Z
Y
X
Ex. 1: Finding Measures of Arcs
and Inscribed Angles
• Find the measure
of the blue arc or
angle.
m

NMP

= ½ m NP
N
100°
M
P
½ (100°) = 50°
Ex. 2: Comparing Measures of
Inscribed Angles
A
• Find mACB,
mADB, and
mAEB.
E
The measure of each
angle is half the
measure of AB
m AB = 60°, so the
measure of each
angle is 30°

B

D
C
Corollaries to Th. 12-9
A
• If two inscribed
angles of a circle
intercept the
same arc, then
the angles are
congruent.
• C  D
D
B
C
Ex. 3: Finding the Measure of an
Angle
G
• It is given that
mE = 75°. What
is mF?

• E and F both
intercept GH , so
E  F. So,
mF = mE = 75°
E
75°
F
H
• A quadrilateral can be
inscribed in a circle if
and only if its opposite
angles are
supplementary.
• D, E, F, and G lie on
some circle, C, if and
only if mD + mF =
180° and mE + mG =
180°
F
E
C
G
D
D
• Find the value of
each variable.
• DEFG is inscribed in
a circle, so opposite
angles are
supplementary.
• mD + mF = 180°
• z + 80 = 180
• z = 100
E
z°
120°
80°
y°
G
F
Ex. 5:
D
• Find the value of
each variable.
• DEFG is inscribed in
a circle, so opposite
angles are
supplementary.
• mE + mG = 180°
• y + 120 = 180
• y = 60
E
z°
120°
80°
y°
G
F
• If a right triangle is inscribed in a circle, then the
hypotenuse is a diameter of the circle. Conversely,
if one side of an inscribed triangle is a diameter of
the circle, then the triangle is a right triangle and
the angle opposite the diameter is the right angle.
• B is a right angle if and only if AC is a diameter
of the circle.
A
B
C
B
• Find the value of
each variable.
• AB is a diameter.
So, C is a right
angle and mC =
90°
• 2x° = 90°
• x = 45
Q
A
2x°
C
Find each measure.
mEFH
= 65°