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 10B Problem 4 — Solution

2007 AMC 10B Problem 4

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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 $\angle ABC?$

$35$

$40$

$45$

$50$

$60$

Answer: D

Concepts:circumcircle, circumcenter, and circumradius isosceles triangle angle chasing

Difficulty rating: 1030

Solution:

Since $OA=OB=OC,$ triangles $AOB, BOC,$ and $COA$ are isosceles. The base angles give $\angle ABO=\dfrac{180^\circ-140^\circ}{2}=20^\circ$ and $\angle OBC=\dfrac{180^\circ-120^\circ}{2}=30^\circ.$

Therefore $\angle ABC=20^\circ+30^\circ=50^\circ.$

Thus, the correct answer is D.

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