Question

given: \\( \overline{qr} \cong \overline{st} \\), \\( \triangle qts \\) and \\( \triangle srq \\) are right triangles.

1. name the postulate or theorem you can use to prove \\( \triangle qts \cong \triangle srq \\).

sss postulate
hl theorem
sas postulate
asa postulate

Explanation:

Step1: Recall triangle congruence theorems

• SSS (Side - Side - Side) postulate: If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.
• HL (Hypotenuse - Leg) theorem: In a right - triangle, if the hypotenuse and a leg of one right - triangle are equal to the hypotenuse and a leg of another right - triangle, then the two right - triangles are congruent.
• SAS (Side - Angle - Side) postulate: If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent.
• ASA (Angle - Side - Angle) postulate: If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent.

Step2: Analyze the given right - triangles

• Given that \(\triangle QTS\) and \(\triangle SRQ\) are right - triangles (\(\angle T=\angle R = 90^{\circ}\)).
• We know that \(\overline{QR}\cong\overline{ST}\) (given as one pair of sides).
• Also, \(\overline{QS}\) is the hypotenuse for both right - triangles \(\triangle QTS\) and \(\triangle SRQ\) (common side, so \(\overline{QS}\cong\overline{QS}\)).

Since we have a right - triangle situation where the hypotenuse (\(QS\)) and a leg (\(QR\) and \(ST\)) are congruent, we use the HL (Hypotenuse - Leg) theorem.

Answer:

HL Theorem