Question

一个等腰三角形的两个角分别为 $70^\circ$ 和 $x^\circ$ 。 $x$ 的三个可能值之和是多少？

Two angles of an isosceles triangle measure $70^\circ$ and $x^\circ.$ What is the sum of the three possible values of $x?$

$95$

$125$

$140$

$165$

$180$

所有可能情形如下图所示。

在第一种情形中，由等腰三角形性质可得 $x = 70$ 。

在第二种情形中，有因此 $x = 40$ 。 $70 \cdot 2 + x = 180,$

在第三种情形中，有因此 $x = 55$ 。 $2x + 70 = 180,$

这些值之和为 $70 + 40 + 55 = 165.$

所以正确答案是 D。

All the following possibilities are shown below.

In the first scenario, we get $x = 70$ by the properties of the isosceles triangle.

In the second scenario, we get that $70 \cdot 2 + x = 180,$ from which we get that $x = 40.$

From the third scenario, we get that $2x + 70 = 180,$ from which we get that $x = 55.$

The sum of these values yields $70 + 40 + 55 = 165.$

Thus, D is the correct answer.