Question

If $D$ and $E$ are the mid-points of the sides $AB$ and $AC$ respectively of $△ABC$. $DE$ is produced to $F$. To prove that $CF$ is equal and parallel to $DA$, we need an additional information which is:

• A. $△DAE$ = $△EFC$
• B. $AE=EF$
• C. $DE=EF$
• D. $△ADE=△ECF$

Answer 1

The correct option is C $DE=EF$

Given $CF$ is equal and parallel to $DA$

$D$ is the midpoint of $AB$

So, $AD$ =$DB$= $CF$= $\frac{c}{2}$

Similarly $AE$ =$EC$=$\frac{b}{2}$

Let $\angle AED=\alpha$ $ADE=\beta$and$DAE=\gamma$

Since $AD$ is parallel to $FC$

$\angle FEC=\alpha \angle EFC=\beta$and$\angle ECF=\gamma$

Applying $S.A.S$ Congruency for triangle$ADE$ and $CFE$, $AD=CF$, $AE=AC$

$\angle DAE=\angle ECF$

So ${\Delta }^{k}DAE$ and${\Delta }^{k}FCE$ are congruent

So $DE=EF$

It is given that $DE=EF$,then by $S.S.S.$ congruency all angles will be equal and $CF$ and parallel to $DA$

$\therefore$ Additionsl information needed is $DE=EF$.