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Question

# In the above figure, ABCD is a square and BCE is an equilateral triangle, what is the measure of angle DEC?

A
15
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B
30
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C
45
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D
20
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Solution

## The correct option is A 15∘Given ABCD is a square and BCE is an equilateral triangle ∠BCD=90∘ [Interior angle of a square] ∠BCE=60∘ [Interior angle of an equilateral triangle] ∴∠DCE=90∘+60∘=150∘ BC = CD (Sides of a square) ------ (1) BC = CE (Sides of an equilateral triangle) ------ (2). From (1) and (2), DC = EC In ΔDCE, DC = CE ⇒∠CDE=∠CED --------- (3) [Angles opposite equal sides] Also, In ΔDCE ∠DCE+∠CDE+∠CED=180∘ --------- (4) [Angle sum property] From (3) and (4), 2∠DEC+∠CED=180∘ ⇒2∠DEC=180∘−150∘=30∘ ∠DEC=12(30∘)=15∘

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