Question

Write proofs in two–column form.

Given:$\overline{AD}\parallel \overline{ME};\overline{MD}\parallel \overline{BE}$

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

Short answer

Statement	Reason
1.$\angle DAM\cong \angle EMB$	Corresponding angles
2.$\overline{AM}\cong \overline{MB}$	Definition of midpoint
3.$\angle DMA\cong \angle EBM$	Corresponding angles
4.width="113" height="20" role="math">ΔDAM≅ΔEMB	ASA congruency criteria
5.width="67" height="23" role="math">MD¯≅BE¯	corresponding parts of congruent triangle are congruent

Step-by-step solution

Step 1. Observe from figure.

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

Step 2. Show that ∠DAM≅∠EMB; ∠DMA≅∠EBM.

Since$\angle DAM-\angle EMB$ and$\angle DMA-\angle EBM$ form pairs of corresponding angles

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

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$\Delta DAM\cong \Delta EMB$ by ASA (Angle-Side-Angle) congruency criteria

Step 5. Show that MD¯≅BE¯.

Since, corresponding parts of congruent triangle are congruent

Thus,$\overline{MD}\cong \overline{BE}$