Question

Write proofs in two–column form.

Given: $M$is the midpoint of $\overline{AB}$;$\angle 1\cong \angle 2;\angle 3\cong \angle 4$

Prove: $\overline{AC}\cong \overline{BD}$

Short answer

Two column proof is

Statement	Reason
1.$\angle 1\cong \angle 2$	Given
2.$\overline{AM}\cong \overline{BM}$	Definition of midpoint
3.$\angle 3\cong \angle 4$	Given
4.$\Delta WOX\cong \Delta \text{ZOY}$	ASA congruency criteria
5.$\overline{AC}\cong \overline{BD}$	corresponding parts of congruent triangle are congruent

Step-by-step solution

Step 1. Show that AM¯≅BM¯.

By definition of midpoint

$\overline{AM}\cong \overline{BM}$

Step 2. Show ΔACM≅ΔBDM.

Using step 1 and given statement $\Delta ACM\cong \Delta BDM$ by ASA (Angle-Side-Angle) congruency criteria

Step 3. Show AC¯≅BD¯.

Since, corresponding parts of congruent triangle are congruent

So, $\overline{AC}\cong \overline{BD}$

Thus, two column proof is

Statement	Reason
1.$\angle 1\cong \angle 2$	Given
2.$\overline{AM}\cong \overline{BM}$	Definition of midpoint
3.$\angle 3\cong \angle 4$	Given
4.$\Delta WOX\cong \Delta \text{ZOY}$	ASA congruency criteria
5.$\overline{AC}\cong \overline{BD}$	corresponding parts of congruent triangle are congruent