Points A, B, and C lie on a circle. The measure of minor arc AB is 110^\circ and the measu… — SAT

Points $A$ , $B$ , and $C$ lie on a circle. The measure of minor arc $AB$ is $110^\circ$ and the measure of minor arc $BC$ is $130^\circ$ . What is the measure of the inscribed angle $\angle ABC$ , in degrees?

A
120
B
50
60Answer
D
55

Answer

The measure of the major arc

ABC

is the sum of the minor arcs

AB

and

BC

, which is

110^\circ + 130^\circ = 240^\circ

. The remaining minor arc

AC

has a measure of

360^\circ - 240^\circ = 120^\circ

. By the inscribed angle theorem, the measure of the inscribed angle

\angle ABC

is half the measure of its intercepted arc, minor arc

AC

. Therefore, the measure of

\angle ABC

\frac{120^\circ}{2} = 60^\circ

Step-by-Step Solution

Calculate the measure of the major arc

ABC

by adding the measures of the two adjacent minor arcs

AB

and

BC

The measure of arc

ABC

110^\circ + 130^\circ = 240^\circ

Since points

A

B

, and

C

are in order on the circle, the major arc connecting

A

and

C

through

B

is the sum of the arcs

AB

and

BC

Find the measure of the remaining minor arc

AC

The measure of minor arc

AC

360^\circ - 240^\circ = 120^\circ

A full circle measures

360^\circ

. Subtracting the major arc

ABC

from

360^\circ

yields the measure of the minor arc

AC

Apply the inscribed angle theorem to find the measure of angle

ABC

The measure of angle

ABC

\frac{120^\circ}{2} = 60^\circ

The inscribed angle theorem states that the measure of an inscribed angle is half the measure of its intercepted arc. Angle

ABC

intercepts the minor arc

AC

Key Concept

The measure of an inscribed angle is half the measure of its intercepted arc.

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