Question

点 $O$ 是 $\triangle ABC$ 外接圆的圆心，如图所示， $\angle BOC=120^\circ$ ， $\angle AOB=140^\circ$ ，求 $\angle ABC$ 的度数。

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 $\angle ABC?$

$35$

$40$

$45$

$50$

$60$

答案：D

知识点：圆周角角度和

难度评级：1190

解答：

点 $O$ 周围角和为 $360^\circ$ ，所以 $\begin{aligned} \angle AOC&=360^\circ-140^\circ-120^\circ \\ &=100^\circ. \end{aligned}$

由圆周角定理， $\angle ABC$ 与圆心角 $\angle AOC$ 截同一弧 $AC$ ，所以 $\angle ABC=\tfrac12\angle AOC=50^\circ.$

所以正确答案是 D。

The angles around $O$ sum to $360^\circ,$ so $\begin{aligned} \angle AOC&=360^\circ-140^\circ-120^\circ \\ &=100^\circ. \end{aligned}$

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

Thus, the correct answer is D.

2007 AMC 12B 第 3 题

答案：D

难度评级：1190

解答：

其他年份的第 3 题