Download Measure of an inscribed angle

10.4  Inscribed Angles
 inscribed angle
 vertex is on the circle.
 sides contain chords of the circle.
Intercepted
arc
inscribed
angle
Theorem:
Measure of an inscribed angle
If an angle is inscribed in a circle, then its
measure is half the measure of its
intercepted arc.
Intercepted
arc = 64°
inscribed
angle
32°
66°
66°
54°
43°
86°
43°
Theorem: If two inscribed angles of a
circle intercept the same arc, then the
angles are congruent.
Intercepted
arc = 64°
inscribed
angle
32°
32°
Find the measure in circle O:
74°
Find the measure in circle O:
132°
360
 96
264  2  132
96°
Find the measure in circle O:
43.5°
Find the measure in circle O:
56°
Find the measure in circle O:
19°
38°
142°
Find the measure in circle O:
21°
138°
42°
If all the vertices of a polygon lie on a circle:
 the polygon is inscribed in the circle,
and
 the circle is circumscribed about the
polygon.
inscribed
Theorem
A quadrilateral can be inscribed in a circle if
and only if its opposite angles are
supplementary.
mA  mC  180
A
B
mB  mD  180
D
C
EXAMPLE 5
Use Theorem 10.10
Find the value of each variable.
a.
SOLUTION
a.
PQRS is inscribed in a circle, so opposite angles
are supplementary.
m
P+m
R = 180o
75o + yo = 180o
y = 105
m
Q+m
S = 180o
80o + xo = 180o
x = 100
EXAMPLE 5
Use Theorem 10.10
Find the value of each variable.
b.
SOLUTION
b.
JKLM is inscribed in a circle, so opposite angles
are supplementary.
m K + m M = 180o
m J + m L = 180o
2ao + 2ao = 180o
4bo + 2bo = 180o
4a = 180
6b = 180
a = 45
b = 30
Assignment
Page 676-677
3-26 all, 28, 30