The point O is the center of the circle circumscribed about △ ABC, with ∠ BOC=120° and ∠ AOB=140°, as shown. What is the degree measure of ∠ ABC?

2007 AMC 12B Problem 3

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The point

O

is the center of the circle circumscribed about

\triangle ABC,

with

\angle BOC=120^\circ

and

\angle AOB=140^\circ,

as shown. What is the degree measure of

\angle ABC?

35

40

45

50

60

The three central angles at

O

add up to

360^\circ

The inscribed angle

\angle ABC

is half of the central angle

\angle AOC

The angles around

O

sum to

360^\circ,

\begin{aligned} \angle AOC&=360^\circ-140^\circ-120^\circ \\ &=100^\circ. \end{aligned}

By the inscribed angle theorem,

\angle ABC

subtends the same arc

AC

as the central angle

\angle AOC,

\angle ABC=\tfrac12\angle AOC=50^\circ.

Thus, the correct answer is D.