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10.4 Inscribed Angles
Geometry
Spring 2011
Objectives/Assignment
• Use inscribed angles to solve
problems.
• Use properties of inscribed
polygons.
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 10.8: 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
Theorem 10.9
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
Ex. 4: Using the Measure of an
Inscribed Angle
• Theater Design.
When you go to the
movies, you want to
be close to the
movie screen, but
you don’t want to
have to move your
eyes too much to
see the edges of the
picture.
Ex. 4: Using the Measure of an
Inscribed Angle
• If E and G are the
ends of the
screen and you
are at F, mEFG
is called your
viewing angle.
Ex. 4: Using the Measure of an
Inscribed Angle
• You decide that
the middle of the
sixth row has the
best viewing
angle. If
someone else is
sitting there,
where else can
you sit to have
the same viewing
angle?
Ex. 4: Using the Measure of an
Inscribed Angle
• Solution: Draw
the circle that is
determined by
the endpoints of
the screen and
the sixth row
center seat. Any
other location on
the circle will
have the same
viewing angle.
Using Properties of Inscribed
Polygons
• If all of the vertices
of a polygon lie on a
circle, the polygon is
inscribed in the circle
and the circle is
circumscribed about
the polygon. The
polygon is an
inscribed polygon and
the circle is a
circumscribed circle.
Theorem 10.10
• 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
Theorem 10.11
• 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
Ex. 5: Using Theorems 10.10
and 10.11
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
Ex. 5: Using Theorems 10.10
and 10.11
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
z°
E
120°
80°
y°
G
F
Ex. 5: Using Theorems 10.10
and 10.11
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
z°
E
120°
80°
y°
G
F
Ex. 6: Using an Inscribed
Quadrilateral
A
• In the diagram, ABCD is
inscribed in circle P.
Find the measure of
each angle.
2y°
D
3y°
3x°
B
• ABCD is inscribed in a
2x°
circle, so opposite
C
angles are
supplementary.
To solve this system of linear equations,
• 3x + 3y = 180
you can solve the first equation for y to
• 5x + 2y = 180
get y = 60 – x. Substitute this
expression into the second equation.
Ex. 6: Using an Inscribed
Quadrilateral
•
•
•
•
•
•
5x + 2y = 180.
5x + 2 (60 – x) = 180
5x + 120 – 2x = 180
3x = 60
x = 20
y = 60 – 20 = 40
Write the second equation.
Substitute 60 – x for y.
Distributive Property.
Subtract 120 from both sides.
Divide each side by 3.
Substitute and solve for y.
x = 20 and y = 40, so mA = 80°, mB = 60°,
mC = 100°, and mD = 120°