Download Geometry 10-4 Inscribed Angles

Geometry 10-4 Inscribed Angles
An inscribed angle
is an angle whose vertex is
on the circle and whose sides are chords of the circle. The
intercepted arc
of the inscribed angle is the arc
of the circle between the two chords.
Theorem 10.6 - Inscribed Angle Theorem: If an angle is
inscribed in a circle, then the measure of the angle is equal
to half the measure of the intercepted arc.
C
B
C
Angle C is an inscribed angle.
AB is its intercepted arc.
B
m C = 1/2 mAB
mAB = 2 m C
When the angle is ON the circle,
then the angle is half the arc.
A
A
D
40
C
98
Find mCF.
40 = 1/2 CF
CF = 80
E
Find m C.
F
angle = 1/2 arc
angle = 1/2 arc
Theorem 10.7: If two inscribed angles of a circle intercept
the same arc (or congruent arcs), then the angles are
congruent.
P
Q
S
C = 1/2(98) = 49
Since P and S both intercept QR,
then P ~
= S.
R
If m R = 3x and m Q = x + 16, find m R.
3x = x + 16
x=8
R = 3(8) = 24
2x = 16
Given: QR ~
= ST, PQ ~
= PT
~ PTS
Prove: PQR =
Q
Statements
Reasons P
1. QR ~
= ST, PQ ~
= PT 1. Given
~ SPT
T
2. QPR =
2. Thm 10.7
~ PT
3. PQ =
3. Thm 10.5
4. PQ + QR = PR
4. Arc Add Post
PT + TS = PS
~ PS
5. PR =
5. Substitution
~ PS
6. PR =
6. Thm 10.5
~
7. PQR = PTS
7. SAS
R
S
Theorem 10.9: If a quadrilateral is inscribed in a circle, then
its opposite angles are supplementary.
Y
X 95
60
W
Z
Find m Y and m Z.
Y + 60 = 180
Y = 120
Z + 95 = 180
Z = 85
Theorem 10.8: An inscribed angle of a triangle intercepts a
diameter or a semicircle if and only if the angle is a right
angle.
m M = 7x + 2; m P = 17x - 8
N
Find m M.
M
7x + 2 + 90 + 17x - 8 = 180
P
24x + 84 = 180
24x = 96
Q
x=4
m M = 7(4) + 2 = 30