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

2007 AMC 12B Problem 7 — Solution

2007 AMC 12B Problem 7

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

$90$

$108$

$120$

$144$

$150$

Answer: E

Concepts:equilateral triangle square (geometry)angle sum

Difficulty rating: 1330

Solution:

Because $AB=BC=EA$ and $\angle A=\angle B=90^\circ,$ quadrilateral $ABCE$ is a square, so $\angle AEC=90^\circ$ and $EC$ equals the common side length.

Then $CD=DE=EC,$ so $\triangle CDE$ is equilateral and $\angle CED=60^\circ.$ Therefore $\begin{aligned} \angle E&=\angle AEC+\angle CED \\ &=90^\circ+60^\circ \\ &=150^\circ. \end{aligned}$

Thus, the correct answer is E.

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