In the diagram shown, ^@ ABCD ^@ is a square and point ^@ F ^@ lies on ^@ BC. ^@ ^@ \triangle DEC ^@ is equilateral and ^@ EB = EF. ^@ What is the measure of ^@ \angle EBC ? ^@
D C B A E F


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Answer:

^@ 75^\circ ^@

Step by Step Explanation:
  1. Given, ^@ ABCD ^@ is a square, ^@ \triangle DEC ^@ is an equilateral triangle and ^@ EB =EF. ^@
    ^@ \implies \angle DCB = 90^\circ \text{ and } \angle DCE = 60^\circ ^@
    ^@ \implies \angle ECF = 30^\circ ^@
  2. Since ^@ DC = CE \space\space\space\space ^@ [Sides of an equilateral triangle]
    and ^@ DC = CB \space\space\space\space ^@ [Sides of a square]
    ^@ \implies CE = CB ^@
    ^@\implies \triangle ECB ^@ is an isosceles triangle.
    ^@\implies \angle EBC = \angle BEC\space\space\space\space\space\space\space\space\space\space\space\space [\because \text{Angles opposite to equal sides of a triangle are equal}] \space\space\space\space^@
    Now, ^@\angle ECB + \angle EBC + \angle BEC = 180^\circ \space\space\space\space\space\space\space\space\space\space\space\space [\text{ Angle Sum Property of a triangle}]^@
    ^@\implies \angle EBC + \angle EBC + 30^\circ = 180^\circ^@
    ^@\implies \angle EBC = \dfrac{ (180 - 30) } { 2 } = 75^\circ ^@
  3. Given, ^@ EB = EF ^@
    ^@ \therefore \angle BFE = \angle EBC = 75^\circ ^@
  4. Hence, the value of ^@ \angle EBC ^@ is ^@ \angle 75^\circ. ^@

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