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Section 11.5
Inscribed Angles and Polygons
INSCRIBED ANGLE:
INTERCEPTED
Ex.
is an angle whose vertex is on a circle and whose sides
contain chords of the circle.
ARC: the arc that is formed by an inscribed angle.
Inscribed angle: ______
Intercepted Arc: ______
A
Theorem 11.7
If an angle is inscribed in a circle,
then the measure of the angle equals
one-half the measure of its intercepted arc.
E
C
B
D
m BAC __________.
Find the measure of its inscribed angle or the intercepted arc.
1.
2.
B
N
90
P
A
C
120
K
3. If mST = 68°, find m 1 and m 2.
4. In
M
F, mWX = 20°, mXY = 40°, mUZ =108°,
and mUW = mYZ. Find the measures of the
numbered angles.
R
S
1
W
X
3
Q
2
P
Y
4
T
5
F
2
U
1
Z
INSCRIBED POLYGON: a polygon whose vertices all lie on a circle. The circle is
circumscribed about the polygon and is called a CIRCUMSCRIBED circle.
Inscribed
Triangle
Inscribed
Quadrilateral
circumscribed
circle
Theorem 11.8
A
If a triangle inscribed in a circle is a
right triangle, then the hypotenuse is
a diameter of the circle.
D
If a side of a triangle inscribed in a
circle is a diameter of the circle, then
the triangle is a right triangle.
C
B
If ABC is a right triangle, then AC is a __________________ and m ABC _____.
If AC is a diameter, then ABC is a ______________________.
Find the value of x and y in circle C.
5.
6.
x
y
35
x
C
y
C
x
7. In X. m 4 = 7x + 3, m 5 = 7x + 3, and m 1 = 5x. Find m 1, m 2, m 3, m 4, m 5,
and m 6.
8. Right triangles TVU and TSU are inscribed in P with VU
each numbered angle if m 2 = x + 9 and m 4 = 2x + 6.
SU . Find the measure of
D
Theorem 11.9
If a quadrilateral is inscribed in a circle,
then its opposite angles are supplementary.
A
P
If the opposite angles of a quadrilateral are
supplementary, then the quadrilateral can be
inscribed in a circle.
C
B
Find the values of x and y in circle C.
9.
10.
y
x
y
C
C
x
50
80
100
95
11. Quadrilateral QRST is inscribed in circle M. If m Q=(3y)° , m R=(5x-10)°, m S=(y+20)°
and m T=(3x-18)°, find m QRS and m TQR.
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