Prove that triangle EDM is an isosceles right angel triangle

Question

In the diagram below triangle ABE is an isosceles right angle triangle and triangle ADC is an isosceles right angel triangle and M is in the middle of BC prove that triangle EMD is an isosceles right angel triangle

Answer 1

Let $L$ be a midpoint of $AB$, and $K$ be a midpoint of $AC$.

$EL=\frac{1}{2}AB=MK$

$ML=\frac{1}{2}AC=DK$

$\angle ELM=\angle BLM + 90^{\circ}=\angle MKC+90^{\circ}=\angle MKD$

By SAS congruence criterion $\triangle ELM\equiv \triangle MKD$

From congruence we get $EM=DM$ and $\angle EML=\angle MDK$.

$\angle EMD=\angle EML+\angle LMK+\angle KMD=\angle MDK+ \angle MKC+\angle KMD=180^{\circ}-\angle CKD=90^{\circ}$