Question

Write proofs in two-column form.

Given:$\overline{CD}\cong \overline{AB};\overline{CB}\cong \overline{AD}$

Prove:$\angle 1\cong \angle 2$

Short answer

Statement	Reason
1.$\overline{CD}\cong \overline{AB}$	Given
2.$\overline{BD}\cong \overline{BD}$	Common line segment
3.$\overline{CB}\cong \overline{AD}$	Given
4. $\mathrm{\Delta }CBD\cong \mathrm{\Delta }ADB$	ASA congruency criteria
5.$\angle 1\cong \angle 2$	Corresponding parts of congruent triangle

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 ΔADB≅ΔCBD.

From step 1 and given

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

Step 3. Show that ∠1≅∠2.

Since, corresponding parts of congruent triangle are congruent

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

Thus, two column proof is

Statement	Reason
1.$\overline{CD}\cong \overline{AB}$	Given
2.$\overline{BD}\cong \overline{BD}$	Common line segment
3.$\overline{CB}\cong \overline{AD}$	Given
4.localid="1649574757985" $\mathrm{\Delta }CBD\cong \mathrm{\Delta }ADB$	ASA congruency criteria
5.$\angle 1\cong \angle 2$	Corresponding parts of congruent triangle