Question

Tell which pairs of congruent parts and what methods $\mathbf{(}\mathit{S}\mathit{S}\mathit{S}\mathbf{,}\mathbf{}\mathit{S}\mathit{A}\mathit{S}\mathbf{,}\mathbf{}\mathit{A}\mathit{S}\mathit{A}\mathbf{,}\mathbf{}\mathit{A}\mathit{A}\mathit{S}\mathbf{,}\mathbf{}\mathbf{or}\mathbf{}\mathit{H}\mathit{L}\mathbf{)}$you would use to prove the triangles are congruent.

Given: $\angle \mathbf{3}\cong \angle \mathbf{4}\mathbf{;}\mathbf{}\mathbf{\angle }\mathbf{5}\mathbf{\cong }\mathbf{\angle }\mathbf{6}$

$△\mathbf{PQX}\cong △\mathbf{PTY}$by what method?

Short answer

The congruent parts in the triangles$△PQX$ and$△PTY$ are:

$\angle \mathbf{PQX}\cong \angle \mathbf{PTY}\mathbf{,}\mathbf{}\mathbf{\angle }\mathbf{PXQ}\mathbf{\cong }\mathbf{\angle }\mathbf{PYT}$ and $\mathbf{PX}\cong \mathbf{PY}$.

The triangles$△PQX$ and$△PTY$ are congruent by $\mathit{A}\mathit{A}\mathit{S}$postulate.

Step-by-step solution

- Observe the given diagram.

The given diagram is:

- Write the theorem 4-2.

The theorem $\mathbf{4}\mathbf{-}\mathbf{2}$states that if two angles of a triangle are congruent, then the sides opposite to those angles are congruent.

- Description of step.

As, $\angle 3\cong \angle 4$,therefore it can be said that $\angle PQX\cong \angle PTY$.

As, $\angle 5\cong \angle 6$, therefore by using the theorem $4-2$, it can be said that sides $PX$and $PY$are congruent sides.

Therefore, $PX\cong PY$.

From the given figure it can be noticed that the angles $\angle PXQ$and $\angle PXY$lies on the same straight line.

Therefore, the sum of the angles $\angle PXQ$and $\angle PXY$is role="math" localid="1648896543536" $180°$.

Therefore, $m\angle PXQ+m\angle PXY=180°$.

Now it can also be noticed that:

$\begin{array}{l}m\angle PXQ+m\angle PXY=180°\\ m\angle PXQ+\angle 5=180°\\ m\angle PXQ=180°-\angle 5\end{array}$

From the given figure it can be noticed that the angles $\angle PYT$and $\angle PYX$lies on the same straight line.

Therefore, the sum of the angles $\angle PYT$and $\angle PYX$is $180°$.

Therefore, $m\angle PYT+m\angle PYX=180°$.

Now it can also be noticed that:

$\begin{array}{l}m\angle PYT+m\angle PYX=180°\\ m\angle PYT+\angle 6=180°\\ m\angle PYT=180°-\angle 6\\ m\angle PYT=180°-\angle 5(\because \angle 5=\angle 6)\end{array}$

Therefore, it can be noticed that $m\angle PXQ=m\angle PYT$.

That implies, $\angle PXQ\cong \angle PYT$

In the triangles $△PQX$and $△PTY$, it can be noticed that $\angle PQX\cong \angle PTY,\angle PXQ\cong \angle PYT$and $PX\cong PY$.

Therefore, the triangles $△PQX$and $△PTY$are the congruent triangles by using the $AAS$postulate.

- Write the conclusion.

The postulate that proves the triangles$△PQX$ and$△PTY$ are congruent is $AAS$ postulate.