Question

Question 146

$\Delta DEF$ and $\Delta LMN$ are both isosceles with DE = DF and LM = LN, respectively. If DE = LM and EF = MN, then are the two triangles congruent? Which condition do you use?
If $\angle E={40}^{\circ }$, what is the measure of $\angle N$?

Answer 1

Here,
DE = DF....(i)
LM = LN...(ii)
DE = LM...(iii)

From Eqs. (i), (ii) and (iii), we get DE = DF = LM = LN
In $\Delta DEF$ and $\Delta LMN$, DE = LM [given]
EF = MN [given]
DF = LN [proved above]
By SSS congruence criterion, $\Delta DEF\cong \Delta LMN$
$\therefore$ $\angle E=\angle M$ [by CPCT]

$\angle M={40}^{\circ }$
Also, $\angle M=\angle N$
[because,angles opposite to equal sides are equal]
$\Rightarrow \angle N={40}^{\circ }$