Question

ABCD is a parallelogram. The sides AB and AD are produced to E and F respectively, such that $\mathrm{AB}=\mathrm{BE}$ and $\mathrm{AD}=\mathrm{DF}$.Prove that $△\mathrm{BEC}\cong △\mathrm{DCF}$.

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Solution

Step $1:$ Drawing the diagram:ABCD is a parallelogram which means $\mathrm{AD}\parallel \mathrm{BC}$ and $\mathrm{AB}\parallel \mathrm{CD}$.Sides AB and AD are produced to E and F, such that $\mathrm{AB}=\mathrm{BE}$ and $\mathrm{AD}=\mathrm{DF}$.Step $2:$ Proving $△\mathrm{BEC}\cong △\mathrm{DCF}$:Corresponding angles for $\mathrm{AB}\parallel \mathrm{CD}$,$\angle \mathrm{BAD}=\angle \mathrm{CDF}$Corresponding angles for $\mathrm{AD}\parallel \mathrm{BC}$,$\angle \mathrm{BAD}=\angle \mathrm{CBE}$Therefore, $\angle \mathrm{CDF}=\angle \mathrm{CBE}$……………………….$\left(1\right)$ABCD is a parallelogram, so we know that,$\mathrm{AD}=\mathrm{BC}$ and $\mathrm{AB}=\mathrm{CD}$……….(opposite sides of parallelogram are equal)Given,$\mathrm{AB}=\mathrm{BE}$ and $\mathrm{AD}=\mathrm{DF}$$\therefore \mathrm{DF}=\mathrm{BC}$…………………….. $\left(2\right)$Similarly,$\therefore \mathrm{CD}=\mathrm{BE}$…………………………$\left(3\right)$In, $△\mathrm{DCF}$ and $△\mathrm{BEC}$$\begin{array}{rcl}\mathrm{CD}& =& \mathrm{EB}\mathrm{From}\mathrm{equation}\left(3\right)\\ \angle \mathrm{CDF}& =& \angle \mathrm{EBC}\mathrm{From}\mathrm{equation}\left(1\right)\\ \mathrm{DF}& =& \mathrm{BC}\mathrm{From}\mathrm{equation}\left(2\right)\end{array}$Therefore, by SAS congruency criteria,$△\mathrm{BEC}\cong △\mathrm{DCF}$Hence proved.

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