Question

Given $▱AZBY;\overline{AY}\cong \overline{BX}$. Prove $\angle 1\cong \angle 2,;\angle 1\cong \angle 3$.

Short answer

$\angle 1\cong \angle 2,;\angle 1\cong \angle 3$

Step-by-step solution

Step 1. Check the figure.

Consider the figure.

Step 2. Step description.

The opposite sides of the rhombus are parallel am transversal.

Step 3. Step description.

From the figure, $\overline{ZY}\cong \overline{BX}$.

From the triangles $\Delta AYZ$and $\Delta ABZ$,

Sides of rhombus are equal thus, $ZB=AB$.

Since the angles opposite to equal sides are equal thus $\angle 1\cong \angle 3$ …... (1)

Since the opposite sides of a rhombus are parallel and transversal $AZ$thus $AY\parallel BZ$.

As the alternate angles are congruent thus, $\angle 1\cong \angle 4$ …… (2)

Therefore, from the equations (1) and (2) it follows that $\angle 1\cong \angle 2$.

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