Proving Congruent Triangles

We are asked to name two triangles that are congruent by ASA.

To do so, let's consider two of them at a time.

$△PRQ$ and $△UTS$

In these triangles, we have two pairs of congruent angles, $\angle P$ $\cong$ $\angle U$ and $\angle R$ $\cong$ $\angle T.$ Also, there is a pair of congruent sides, $\overline{PR}$ $\cong$ $\overline{ST}.$ In $△PRQ,$ the congruent side is included between the congruent angles. In $△UTS,$ the congruent side is not included between the congruent angles.

To use ASA, we need two pairs of congruent angles and a pair of congruent included sides. Therefore, we cannot conclude the triangles are congruent by ASA.

$△PRQ$ and $△VWX$

Here we have two pairs of congruent angles, $\angle P$ $\cong$ $\angle V$ and $\angle R$ $\cong$ $\angle W$. Also, there is a pair of congruent sides, $\overline{PR}\cong \overline{VW}.$

We see that in both triangles the congruent sides are included between the congruent angles. Therefore, $△PRQ$ $\cong$ $△VWX$ by ASA.

$△SUT$ and $△XVW$

In these triangles we have two pairs of congruent angles, $\angle U$ $\cong$ $\angle V$ and $\angle T$ $\cong$ $\angle W.$ Also, there is a pair of congruent sides, $\overline{ST}$ $\cong$ $\overline{VW}.$

In $△SUT,$ the congruent side is not included between the congruent angles. In $△XVW,$ the congruent side is included between the congruent angles. Therefore, we cannot conclude the triangles are congruent by ASA.