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

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

The sides

AB,BC,EA

and the two right angles make

ABCE

a square

Then

\triangle CDE

is equilateral, and

\angle E=\angle AEC+\angle CED

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,

\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.