INSCRIBED ANGLESGeometry H2 (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

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

Example—Inscribed Angle

Find the values of a and b. 32

b

a

Example—Inscribed angles P a

Find the values of a and b. T 30 60 S Q 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 arcsso, <A <D

Theorem 12-4-3

An angle inscribed in a semicircle is a right angle.

M

N P

O

< M has an intercepted arc of

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 D

Example—Inscribed Quadrilateral E

Given m = 70, m = 80, and m 90. Find m < G and m <D.

D F

G