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

2007 AMC 12B Problem 3 — Solution

2007 AMC 12B Problem 3

Below is the professionally curated solution for Problem 3 of the 2007 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2007 AMC 12B solutions, or check the answer key.

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Concepts:inscribed angle angle sum

Difficulty rating: 1190

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 $\angle ABC?$

$35$

$40$

$45$

$50$

$60$

Solution:

The angles around $O$ sum to $360^\circ,$ so $\angle AOC=360^\circ-140^\circ-120^\circ=100^\circ.$

By the inscribed angle theorem, $\angle ABC$ subtends the same arc $AC$ as the central angle $\angle AOC,$ so $\angle ABC=\tfrac12\angle AOC=50^\circ.$

Thus, the correct answer is D.

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