Question

角 $A$ 的度数是

The degree measure of angle $A$ is

$20$

$30$

$35$

$40$

$45$

答案：B

知识点：导角

难度评级：1520

小提示：

在两个交点处使用补角关系

Use supplementary angles at the two intersections

大提示：

再在靠近 $40^\circ$ 角的三角形中使用三角形内角和

Then use a triangle angle sum near the $40^\circ$ angle

解答：

如下标记顶点。

由补角关系， $\angle ABC = 180^{\circ} - 100^{\circ} = 80^{\circ}\text{。}$ 同样由补角关系， $\angle CED = 180^{\circ} - 110^{\circ} = 70^{\circ}\text{。}$ 利用三角形内角和为 $180^{\circ}$ ，得到 $\begin{aligned} \angle ECD &= 180^{\circ} - 70^{\circ} - 40^{\circ} \\ &= 70^{\circ} \end{aligned}\text{。}$ 再由对顶角相等， $\angle ACB = \angle ECD = 70^{\circ}\text{。}$ 最后在三角形中使用内角和，得到 $\angle A = 180^{\circ} - 80^{\circ} - 70^{\circ} = 30^{\circ}\text{。}$

所以正确答案是 B。

Label the vertices as below.

We then have that $\angle ABC = 180^{\circ} - 100^{\circ} = 80^{\circ}$ by supplementary angles. Then we have $\angle CED = 180^{\circ} - 110^{\circ} = 70^{\circ}$ again by supplementary angles. Using the sum of the interior angles of a triangle is $180^{\circ},$ we get $\begin{aligned} \angle ECD &= 180^{\circ} - 70^{\circ} - 40^{\circ} \\ &= 70^{\circ}. \end{aligned}$ Then, using vertical angles, we have $\angle ACB = \angle ECD = 70^{\circ}.$ Finally, using the sum of the interior angles of a triangle, we get $\angle A = 180^{\circ} - 80^{\circ} - 70^{\circ} = 30^{\circ}.$

Thus, B is the correct answer.

1999 AMC 8 第 21 题

答案：B

难度评级：1520

小提示：

大提示：

解答：

其他年份的第 21 题