In the diagram shown, ABCD is a square and point F lies on BC. △DEC is equilateral and EB=EF. What is the measure of ∠EBC?
Answer:
75∘
- Given, ABCD is a square, △DEC is an equilateral triangle and EB=EF.
⟹∠DCB=90∘ and ∠DCE=60∘
⟹∠ECF=30∘ - Since DC=CE [Sides of an equilateral triangle]
and DC=CB [Sides of a square]
⟹CE=CB
⟹△ECB is an isosceles triangle.
⟹∠EBC=∠BEC [∵Angles opposite to equal sides of a triangle are equal]
Now, ∠ECB+∠EBC+∠BEC=180∘ [ Angle Sum Property of a triangle]
⟹∠EBC+∠EBC+30∘=180∘
⟹∠EBC=(180−30)2=75∘ - Given, EB=EF
∴∠BFE=∠EBC=75∘ - Hence, the value of ∠EBC is ∠75∘.