Question

Given: $▱\text{ABZY}$; $\overline{\text{ZY}}\cong \overline{\text{BX}}$; $\angle 1\cong \angle 2$

Prove: ABZY is a rhombus.

Short answer

The two-column proof of $\text{ABZY}$is a rhombus is:

Statement	Reason
$\overline{\text{ZY}}\cong \overline{\text{BX}}$	Given.
$\angle \mathbf{1}\cong \angle \mathbf{2}$	Given.
$\angle \text{AXB}\cong \angle \text{YXZ}$	Vertical angles are congruent.
$\Delta \text{AXB}\cong \Delta \text{YXZ}$	AAS.
$\text{AX}\cong \text{XZ}$	CPCTC.
$\text{BX}\cong \text{XY}$	CPCTC.
Hence $\angle \text{YAB}$is a right angle.	Definition of a parallelogram.
$\overline{\text{AZ}}$is perpendicular bisector to $\overline{\text{BY}}$and $\overline{\text{BY}}$ is perpendicular bisector to $\overline{AZ}$.	Definition of perpendicular bisector.
$ABZY$is a rhombus.	Definition of a rhombus.

Step-by-step solution

Step 1. Consider the diagram.

Parallelogram $ABZY$is:

Here, $\overline{ZY}\cong \overline{BX}$and $\angle 1\cong \angle 2$.

Step 2. State the proof.

Statement	Reason
$\overline{ZY}\cong \overline{BX}$	Given.
$\angle 1\cong \angle 2$	Given.
$\angle AXB\cong \angle YXZ$	Vertical angles are congruent.
$\Delta AXB\cong \Delta YXZ$	AAS.
$AX\cong XZ$	CPCTC.
$BX\cong XY$	CPCTC.
Hence $\angle YAB$is a right angle.	Definition of a parallelogram.
$\overline{AZ}$is perpendicular bisector to $\overline{BY}$and $\overline{BY}$ is perpendicular bisector to $\overline{AZ}$.	Definition of perpendicular bisector.
$ABZY$is a rhombus.	Definition of a rhombus.

Step 3. State the conclusion.

Therefore, $ABZY$is a rhombus.