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2007 AMC 10B Problem 7

Problem 7 of 25EasierGeometry

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

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Solution

Because AB=BC=EAAB=BC=EA and ∠A=∠B=90∘,\angle A=\angle B=90^\circ, quadrilateral ABCEABCE is a square, so ∠AEC=90∘.\angle AEC=90^\circ. The remaining sides satisfy CD=DE=EC,CD=DE=EC, so △CDE\triangle CDE is equilateral and ∠CED=60∘.\angle CED=60^\circ. Therefore ∠E=∠AEC\angle E=\angle AEC +∠CED=90∘+\angle CED=90^\circ +60∘=150∘.+60^\circ=150^\circ. Thus, the correct answer is E.

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Concepts: angle sum · square (geometry) · equilateral triangle

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.