Chapter 4.3, Problem 8WE

To determine

Toprove: $\overline{MD}\left|\right|\overline{BE}$.

Explanation of Solution

Given information :

If $M$ is the midpoint of $\overline{AB}$ ; $\overline{AD}\left|\right|\overline{ME};\overline{AD}\cong \overline{ME}$.

  

Proof: Since $M$ is the midpoint of $\overline{AB}$ ;

  $\Rightarrow \overline{AM}\cong \overline{BM}$ by the definition of midpoint and also given $\overline{AD}\cong \overline{ME};\overline{AD}\left|\right|\overline{ME};$

If two parallel lines are cut by a transversal then alternate interior angles are congruent.

  $\Rightarrow \angle DAM=\angle EMB$

By SAS theorem, the given two triangles are congruent.

Thus, $\Delta AMD\cong \Delta MBE$

  $\Rightarrow$ Thepairs of sides and angles of two triangles are equal in measurement.

Thus, $\angle AMD=\angle MBE$

  $\text{Hence,}\overline{MD}\left|\right|\overline{BE}$.