Question

Give the reason for each key step of the proof.

Given: $\mathbf{AX}\cong \mathbf{CY}$; $\angle \mathbf{A}\cong \angle \mathbf{C}$; $\overline{\mathbf{BX}}\perp \overline{\mathbf{AD}}$; $\overline{\mathbf{DY}}\perp \overline{\mathbf{BC}}$

Prove: $\overline{\mathbf{AD}}\parallel \overline{\mathbf{BC}}$.

1. $\Delta \mathbf{ABX}\cong \Delta \mathbf{CDY}$.

2. $\overline{\mathbf{BX}}\cong \overline{\mathbf{DY}}$

3. $\Delta \mathbf{BDX}\cong \Delta \mathbf{DBY}$.

4. $\angle \mathbf{1}\cong \angle \mathbf{2}$

5. $\overline{\mathbf{AD}}\parallel \overline{\mathbf{BC}}$

Short answer

The reason for each key step of the proof is:

Step-by-step solution

Step 1.  Observe the given figure.

The given figure is:

Step 2.  Description of step.

It is being given that $AX\cong CY$, $\angle A\cong \angle C$, $\overline{BX}\perp \overline{AD}$and $\overline{DY}\perp \overline{BC}$.

As, $\overline{BX}\perp \overline{AD}$, therefore$\angle AXB=90°$.

As, $\overline{DY}\perp \overline{BC}$, therefore $\angle CYD=90°$.

Therefore, it is obtained that $\angle AXB=\angle CYD$.

Therefore, $\angle AXB\cong \angle CYD$

Therefore, in the triangles $\Delta ABX$and $\Delta CDY$, it can be noticed that$\angle AXB\cong \angle CYD$,$AX\cong CY$and $\angle A\cong \angle C$.

Therefore, $\Delta ABX\cong \Delta CDY$by using ASA postulate.

Therefore, by using the corresponding parts of congruent triangles it can be noticed that $\overline{BX}\cong \overline{DY}$.

In the triangles $\Delta BDX$and $\Delta DBY$, $DB$is the hypotenuse, $\overline{BX}$and $\overline{DY}$are the legs.

Therefore, by using the HL theorem it can be said that $\Delta BDX\cong \Delta DBY$.

Therefore, by using the corresponding parts of congruent triangles it can be noticed that $\angle 1\cong \angle 2$.

If the lines are cut by a transversal then if the alternate interior angles are congruent then lines are parallel.

Therefore, as the alternate interior angles $\angle 1$and $\angle 2$ are congruent, therefore the lines $\overline{AD}$and$\overline{BC}$are parallel.

Step 3.  Description of step.

The reason for each key step of the proof is: