Question

cw: triangle congruence theorems
if not congruent, write none in both slots.
these triangles can be proven congruent by □ △ pqs ≅ □

Explanation:

Step1: Analyze the given information

We have two right - angled triangles \(\triangle PQS\) and \(\triangle RQS\). \(PS = QR\) (given as equal sides), \(QS\) is a common hypotenuse for both right - angled triangles.

Step2: Apply the Hypotenuse - Leg (HL) theorem

In right - angled triangles, if the hypotenuse and one leg of a right - angled triangle are equal to the hypotenuse and one leg of another right - angled triangle, then the two triangles are congruent. Here, for right - angled \(\triangle PQS\) and \(\triangle RQS\), \(QS = QS\) (common hypotenuse) and \(PS = QR\) (leg).

Step3: Determine the congruent triangle

By the HL theorem, \(\triangle PQS\cong\triangle RQS\)

Answer:

These triangles can be proven congruent by \(HL\), \(\triangle PQS\cong\triangle RQS\)