Question

Write proofs in two-column form.

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

Prove:$\overline{DC}\parallel \overline{AB}$

Short answer

Statement	Reason
1.$\overline{CD}\cong \overline{AB}$	Given
2.$\angle 3\cong \angle 4$	Alternate interior angles
3.$\overline{BD}\cong \overline{BD}$	Common line segment
4. $\mathrm{\Delta }CBD\cong \mathrm{\Delta }ADB$	SAS congruency criteria
5.$\angle 1\cong \angle 2$	Corresponding parts of congruent triangle
6.$\overline{DC}\parallel \overline{AB}$	If alternate interior angles are equal then lines are parallel

Step-by-step solution

Step 1. Show that BD¯≅BD¯.

Line segment $\overline{BD}$is common in both triangles, by reflexive property of congruency $\overline{BD}$is congruent to itself

$\overline{BD}\cong \overline{BD}$

Step 2. Show that ∠3≅∠4.

Since$\angle 3$ and$\angle 4$ forms a pair of alternate interior angles

So,$\angle 3\cong \angle 4$

Step 3. Show that ΔCBD≅ΔADB.

From step 1 and 2 and given

$\mathrm{\Delta }CBD\cong \mathrm{\Delta }ADB$by SAS (Side-Angle-Side) congruency criteria

Step 4. Show that ∠1≅∠2.

Since, corresponding parts of congruent triangle are congruent

So,$\angle 1\cong \angle 2$

Step 5. Show that DC¯∥AB¯.

Since,$\angle 1\text{and}\angle 2$ forms a pair of congruent alternate interior angles. So, lines are parallel.

$\overline{DC}\parallel \overline{AB}$

Thus, two column proof is

Statement	Reason
1.$\overline{CD}\cong \overline{AB}$	Given
2.$\angle 3\cong \angle 4$	Alternate interior angles
3.$\overline{BD}\cong \overline{BD}$	Common line segment
4.$\mathrm{\Delta }CBD\cong \mathrm{\Delta }ADB$	SAS congruency criteria
5.$\angle 1\cong \angle 2$	Corresponding parts of congruent triangle
6.$\overline{DC}\parallel \overline{AB}$	If alternate interior angles are equal then lines are parallel