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 ? ^@
Answer:
^@ 75^\circ ^@
- 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 ^@ - 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 ^@ - Given, ^@ EB = EF ^@
^@ \therefore \angle BFE = \angle EBC = 75^\circ ^@ - Hence, the value of ^@ \angle EBC ^@ is ^@ \angle 75^\circ. ^@