Question

Write proofs in two–column form.

Given: $\overline{AD}\cong \overline{ME};\overline{AD}\parallel \overline{ME}$

$M$is the midpoint of $\overline{AB}$. Prove:$\overline{MD}\parallel \overline{BE}$

Short answer

Statement	Reason
1.$\overline{AD}\cong \overline{ME}$	Given
2.$\angle DAM\cong \angle EMB$	Corresponding angles
3.$\overline{AM}\cong \overline{MB}$	Definition of midpoint
4.$\Delta DAM\cong \Delta EMB$	SAS congruency criteria
5.$\angle DMA\cong \angle EBM$	corresponding parts of congruent triangle are congruent
6.$\overline{MD}\parallel \overline{BE}$	When transversal line intersect two lines such that corresponding angles are congruent then lines are parallel.

Step-by-step solution

Step 1. Observe from figure.

Line segment$\overline{AB}$ is transversal to parallel lines$\overline{AD}\parallel \overline{ME}$

Step 2. Show that ∠DAM≅∠EMB.

Since$\angle DAM\text{and}\angle EMB$ forms a pair of corresponding angles

When transversal line intersect two parallel lines then corresponding angles are congruent. Thus,$\angle DAM\cong \angle EMB$

Step 3. Show that AM¯≅MB¯.

By definition of midpoint, midpoint divides line segment in two congruent line segments

$\overline{AM}\cong \overline{MB}$

Step 4. Show that ΔDAM≅ΔEMB.

From step 2 and 3 and using given statement$\mathrm{\Delta }DAM\cong \mathrm{\Delta }EMB$ by SAS (Side-Angle-Side) congruency criteria

Step 5. Show that ∠DMA≅∠EBM.

Since, corresponding parts of congruent triangle are congruent

Thus,$\angle DMA\cong \angle EBM$

Step 6. Show that MD¯∥BE¯.

Since$\angle DMA-\angle EBM$ forms a pair of corresponding angles

When transversal line intersect two lines such that corresponding angles are congruent then lines are parallel. Thus, $\overline{MD}\parallel \overline{BE}$