Inscribed Angles
Section 10.5
Essential Questions
1. How to use inscribed angles to solve
problems in geometry
2. How to use inscribed angles to solve
real-life problems
Definitions
 Inscribed angle – an angle whose vertex is on a circle
and whose sides contain chords of the circle
 Intercepted arc – the arc that lies in the interior of an
inscribed angle and has endpoints on the angle
intercepted arc
inscribed angle
Measure of an Inscribed Angle Theorem (10.9)
 If an angle is inscribed in a circle, then its measure is
half the measure of its intercepted arc.
A
mADB =
1
2
C
mAB
D
B
Example 1
 Find the measure of the blue arc or angle.
a.
S
R
E
b.
80
F
T
G
Q
mQTS = 2(90) = 180
mEFG =
1
2
(80) = 40
Congruent Inscribed Angles Theorem (10.10)
If two inscribed angles of a circle intercept
the same arc, then the angles are
congruent.
A
B
C
D
C  D
Example 2
It is given that mE = 75. What is mF?
Since E and F both intercept
the same arc, we know that the
angles must be congruent.
D
E
mF = 75
F
H
Definitions
 Inscribed polygon – a polygon whose vertices all lie on a
circle.
 Circumscribed circle – A circle with an inscribed polygon.
The polygon is an inscribed polygon and
the circle is a circumscribed circle.
Inscribed Right Triangle Theorem(10.11)
 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.
A
B is a right angle if and only if AC
is a diameter of the circle.
B
C
Inscribed Quadrilateral Theorem(10.12)
 A quadrilateral can be inscribed in a circle if and only if
its opposite angles are supplementary.
E
F
C
D
G
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.
Example 3
 Find the value of each variable.
a.
B
Q
A
D
b.
G
y
F
C
x = 45
E
80
2x
2x = 90
z
120
mD + mF = 180
z + 80 = 180
z = 100
mG + mE = 180
y + 120 = 180
y = 60
Assignment:
P. 508/ 1- 14
HW: p. 508/ 15- 33 all