Download Inscribed Angles

INSCRIBED ANGLES
Geometry CP1 (Holt 12-4)
K. Santos
Inscribed angle
Inscribed angle—an angle in the circle with its vertex on the
circle and its sides being chords of the circle.
A
C
B
< C is and inscribed angle
𝐴𝐵 is the intercepted arc
Inscribed Angle Theorem 12-4-1
The measure of an inscribed angle is half the measure of
its intercepted arc.
X
Y
Z
m < Y = ½ 𝑋𝑍
Example—Inscribed Angle
Find the values of a and b.
32°
b°
a°
a = ½ of a semicircle
a = ½ (180) = 90°
The inscribed angle has an arc 2(32) = 64°
b= 180 – 64
b = 116°
Example—Inscribed angles
P
a°
Find the values of a and b.
T
30°
S
60°
Q
m < PQT = ½ m𝑃𝑇
60 = ½ a
120° = a
m < PRS = ½ m𝑃𝑆
b = ½ (m𝑃𝑇 +m𝑇𝑆)
b = ½ (120 + 30)
b = ½ (150)
b = 75°
b°
R
Corollary to the Inscribed Angle Theorem 12-4-2
Two inscribed angles that intercept the same arc are
congruent.
A
D
B
C
<A and < D have the same intercepted arcs
so, <A ≅<D
Theorem 12-4-3
An angle inscribed in a semicircle is a right angle.
M
P
N
O
< M has an intercepted arc of 𝑁𝑂𝑃
this arc is a semicircle
So, m < M = 90° (1/2 of 180°)
Theorem 12-4-4
If a quadrilateral is inscribed in a circle, then its opposite
angles are supplementary.
A
B
C
D
< A and <C are opposite angles, so they are supplementary
<B and <D are opposite angles, so they are supplementary
Example—Inscribed quadrilateral
m < A = 70° and m < B = 120° . Find m < C and m < D
A
B
C
< A and < C are supplementary,
180 – 70 =110 so m<C = 110°
< B and < D are supplementary
180 – 120 = 60 so m <D = 60°
D
Example—Inscribed Quadrilateral
E
Given m 𝐸𝐹 = 70°, m 𝐷𝐸= 80°, and
m 𝐷𝐺 = 90°. Find m < G and m <D.
D
F
G
Find m 𝐺𝐹
all four arcs add to 360°
70 + 80 + 90 + x = 360
240 + x = 360
m 𝐺𝐹 = 120°
m < G = ½ ( 𝐷𝐸 + 𝐸𝐹 )
m < G = ½ (80 + 70)
m < G = ½ (150)
m < G = 75°
m < D = ½ ( 𝐸𝐹 + 𝐹𝐺 )
m < D = ½ (70 +120 )
m < D = ½ (190)
m < D = 95°