Question

in the accompanying diagram, \\( \triangle abc \\) and \\( \triangle rst \\) are right triangles with right angles at \\( b \\) and \\( s \\), respectively; \\( \overline{ab} \cong \overline{rs} \\) and \\( \overline{ac} \cong \overline{rt} \\).
which statement can be used to justify that \\( \triangle abc \cong \triangle rst \\)?
\\( hl \cong hl \\)
\\( saa \cong saa \\)
\\( asa \cong asa \\)
\\( sas \cong sas \\)

Explanation:

Step1: Recall triangle congruence theorems

For right - angled triangles, the \(HL\) (Hypotenuse - Leg) theorem states that if the hypotenuse and a leg of one right - triangle are congruent to the hypotenuse and a leg of another right - triangle, then the two right - triangles are congruent.
In \(\triangle ABC\) and \(\triangle RST\), \(\angle B=\angle S = 90^{\circ}\) (right angles). We are given that \(\overline{AB}\cong\overline{RS}\) (a leg) and \(\overline{AC}\cong\overline{RT}\) (the hypotenuse).

Step2: Check other congruence theorems

• \(SAA\) (Side - Angle - Angle) is not a standard congruence theorem for right - angled triangles in the context of the given information.
• \(ASA\) (Angle - Side - Angle) requires two angles and the included side. We only know one angle (the right angle) from the given information in a non - included side situation for this pair of triangles.
• \(SAS\) (Side - Angle - Side) requires two sides and the included angle. The right angle is not included between the given side (\(AB\) or \(RS\)) and the hypotenuse (\(AC\) or \(RT\)).

Answer:

\(HL\cong HL\)