Download Geometry Notes 8.7 Inscribed Angles

Geometry Notes
8.7—Inscribed Angles
Objective: Use inscribed angles and properties of inscribed polygons to solve problems.
Inscribed Arc
A
An inscribed angle is an angle whose vertex is on a circle
and whose sides contain chords of the circle.
C
D
An intercepted arc is an arc that lies in the interior of an
inscribed angle and has endpoints on the angle.
B
Inscribed Angle
The measure of an inscribed angle is half the measure of its intercepted arc.
1
mABC  m AC or m AC  2 mABC
2
Theorem 8.4:
If two inscribed angles of a circle intercept the same arc,
then the angles are congruent.
1
2
1  2
Examples:
Find the measure of
1.
86°
 A.
Find the value of each variable.
2.
3.
246°
M
x°
P
H
29°
58°
A
A
y°
W
H
Examples:
R
H
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 8.6:
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
A
C
 B is a right angle if and only if AC is a diameter of the circle.
E
Theorem 8.7:
A quadrilateral can be inscribed in a circle if and only if
its opposite angles are supplementary.
D
mD  mF  180 and mE  mG  180
G
F
Examples:
10.
11.
y°
x°
2y°
3y°
39°
3x°
5x°