Chapter 4.3, Problem 7WE

To determine

Toprove: $\overline{MD}\cong \overline{BE}$.

Explanation of Solution

Given information :

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

  

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

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

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

  $\Rightarrow \angle DAM=\angle EMB,\angle AMD=\angle MBE$

By ASA theorem, the given two triangles are congruent.

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

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

Hence, $\overline{MD}\cong \overline{BE}$.