Question

一个四边形内接于圆。在四边形外部的四个弓形中各作一个圆周角，则这四个角的度数之和为：

A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, the sum of these four angles, expressed in degrees, is:

$1080$

$900$

$720$

$540$

$360$

答案：D

知识点：圆内接四边形圆周角弧角度和

难度评级：1630

小提示：

四边形各边所截的四段弧的度数之和为 $360^\circ$

Let the four arcs cut off by the quadrilateral’s sides have measures summing to $360^\circ$

大提示：

位于某边外侧弓形内的圆周角所对的是互补的优弧

An angle in the outside segment of a side subtends the complementary major arc

解答：

设各边所截的四段劣弧度数为 $\alpha_1,\ldots,\alpha_4$ ，其和为 $360^\circ$ 。与弧 $\alpha_i$ 对应的外侧弓形内的圆周角所对的是另一段弧，因此其度数为 $\frac{360^\circ-\alpha_i}{2}\text{。}$ 所以四个角的度数之和为 $\frac{4(360^\circ)-360^\circ}{2}=540^\circ\text{。}$

所以正确答案为 D。

Let the four minor arcs cut off by the sides have measures $\alpha_1,\ldots,\alpha_4,$ whose sum is $360^\circ.$ The angle in the outside segment corresponding to arc $\alpha_i$ subtends the other arc, so its measure is $\frac{360^\circ-\alpha_i}{2}.$ The sum of the four angles is therefore $\frac{4(360^\circ)-360^\circ}{2}=540^\circ.$

Thus, the correct answer is D.

1958 AMC 12 第 15 题

答案：D

难度评级：1630

小提示：

大提示：

解答：

其他年份的第 15 题