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

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

$△ P R Q$ and $△ U T S$

In these triangles, we have two pairs of congruent angles, $∠ P$ $≅$ $∠ U$ and $∠ R$ $≅$ $∠ T .$ Also, there is a pair of congruent sides, $\overline{P R}$ $≅$ $\overline{S T} .$ In $△ P R Q,$ the congruent side is included between the congruent angles. In $△ U T S,$ 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.

$△ P R Q$ and $△ V W X$

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

We see that in both triangles the congruent sides are included between the congruent angles. Therefore, $△ P R Q$ $≅$ $△ V W X$ by ASA.

$△ S U T$ and $△ X V W$

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

In $△ S U T,$ the congruent side is not included between the congruent angles. In $△ X V W,$ the congruent side is included between the congruent angles. Therefore, we cannot conclude the triangles are congruent by ASA.

△PRQ and △UTS

△PRQ and △VWX

△SUT and △XVW

$△ P R Q$ and $△ U T S$

$△ P R Q$ and $△ V W X$

$△ S U T$ and $△ X V W$

Exercise