Question

Given: $▱\text{ABZY}$; $\overline{\text{AY}}\cong \overline{\text{BX}}$;

Prove: $\angle 1\cong \angle 2$and $\angle 1\cong \angle 3$.

Short answer

$\angle \mathbf{1}\cong \angle \mathbf{2}$and $\angle \mathbf{1}\cong \angle \mathbf{3}$

Step-by-step solution

Step 1. Consider the diagram.

Parallelogram ABZY is:

Here, $\overline{AY}\cong \overline{BX}$.

Step 2. State the proof.

In $\Delta AYZ$and $\Delta ABZ$,

Statement	Reason
$ZB\cong AB$	The sides of a rhombus are equal.
$\angle 1\cong \angle 3$ …… (1)	Angles opposite to equal sides are equal.
$AY\parallel BZ$	The opposite sides of a rhombus are parallel and transversal $AZ$.
$\angle 1\cong \angle 4$ …… (2)	Alternate angles.

From (1) and (2),

$\angle 2\cong \angle 1$

Thus, $\angle 1\cong \angle 2$and $\angle 1\cong \angle 3$.

Step 3. State the conclusion.

Therefore, $\angle 1\cong \angle 2$and $\angle 1\cong \angle 3$ (proved).