AMC10 2007 B

AMC10 2007 B · Q7

AMC10 2007 B · Q7. It mainly tests Triangles (properties), Circle theorems.

All sides of the convex pentagon ABCDE are of equal length, and $\angle A = \angle B = 90^\circ$. What is the degree measure of $\angle E$?

凸五边形ABCDE的所有边长相等，且$\angle A = \angle B = 90^\circ$。$\angle E$的度数是多少？

(A) 90 90

(B) 108 108

(D) 144 144

(E) 150 150

Answer

Correct choice: (E)

正确答案：（E）

Solution

Because AB = BC = EA and $\angle A = \angle B = 90^\circ$, quadrilateral ABCE is a square, so $\angle AEC = 90^\circ$. Also CD = DE = EC, so $\triangle CDE$ is equilateral and $\angle CED = 60^\circ$. Therefore $\angle E = \angle AEC + \angle CED = 90^\circ + 60^\circ = 150^\circ$.

因为AB = BC = EA且$\angle A = \angle B = 90^\circ$，四边形ABCE是正方形，故$\angle AEC = 90^\circ$。同时CD = DE = EC，故$\triangle CDE$是等边三角形，$\angle CED = 60^\circ$。因此$\angle E = \angle AEC + \angle CED = 90^\circ + 60^\circ = 150^\circ$。

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