Question

use the given information to prove that $\triangle pqs \cong \triangle rsq$.
your answer
given: $m \angle 2 = m \angle 3$
$m \angle p = m \angle r$
prove: $\triangle pqs \cong \triangle rsq$
step statement reason
1 $m \angle 2 = m \angle 3$ given
2 $m \angle p = m \angle r$ given
3 $\triangle pqs \cong \triangle rsq$ reason?

Explanation:

Step1: Identify Common Side

Both triangles \(\triangle PQS\) and \(\triangle RSQ\) share the side \(QS\), so \(QS = SQ\) (reflexive property of congruence).

Step2: List Known Angles and Side

We know \(m\angle 2 = m\angle 3\), \(m\angle P = m\angle R\), and \(QS = SQ\).

Step3: Apply AAS Congruence

In \(\triangle PQS\) and \(\triangle RSQ\):

• \(\angle P \cong \angle R\) (given, \(m\angle P = m\angle R\))
• \(\angle 2 \cong \angle 3\) (given, \(m\angle 2 = m\angle 3\))
• \(QS \cong SQ\) (common side, reflexive property)

By the Angle - Angle - Side (AAS) congruence criterion, if two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle, then the two triangles are congruent. So \(\triangle PQS\cong\triangle RSQ\) by AAS.

Answer:

The reason for \(\triangle PQS\cong\triangle RSQ\) is the Angle - Angle - Side (AAS) Congruence Criterion.