Question

marcus and marlee are comparing \\( \triangle a b c \\) with right angle b with \\( \triangle p q r \\) with right angle q to determine if they are congruent. \\( \overline{a b} \cong \overline{q r} \\) and \\( \overline{a c} \cong \overline{p r} \\) marcus says the triangles are congruent by the hl congruence theorem, but marlee does not agree. who is correct? (hint: draw a picture) (1 point) marcus is correct. they are congruent by hypotenuse-leg because both are right triangles where the hypotenuse and a leg of one triangle are congruent to the hypotenuse and a leg of the other triangle. neither is correct. they are congruent by side-side-angle. neither is correct. they are congruent by side-angle-side. marlee is correct. they are not congruent because the corresponding legs are not marked as congruent.

Explanation:

Step1: Recall the HL Congruence Theorem

The HL (Hypotenuse - Leg) Congruence 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 \(\triangle ABC\) (right - angled at \(B\)) and \(\triangle PQR\) (right - angled at \(Q\)), for the HL theorem, the hypotenuse and a leg should be in the correct correspondence. Here, \(\overline{AB}\) is a leg of \(\triangle ABC\) and \(\overline{QR}\) is a leg of \(\triangle PQR\), \(\overline{AC}\) is the hypotenuse of \(\triangle ABC\) and \(\overline{PR}\) is the hypotenuse of \(\triangle PQR\). But for HL, we need the hypotenuse and a corresponding leg (e.g., if we consider \(\triangle ABC\) and \(\triangle PQR\), we should have either \(\overline{AB}\cong\overline{PQ}\) (leg - leg) and \(\overline{AC}\cong\overline{PR}\) (hypotenuse - hypotenuse) or \(\overline{BC}\cong\overline{QR}\) (leg - leg) and \(\overline{AC}\cong\overline{PR}\) (hypotenuse - hypotenuse)).

Step2: Analyze the SSA (Side - Side - Angle) and SAS (Side - Angle - Side)

The SSA (Side - Side - Angle) is not a valid congruence criterion for non - right triangles (and in the wrong correspondence for right - triangles in this case). The SAS (Side - Angle - Side) requires two sides and the included angle to be congruent. Here, we are not given an included angle congruence.

Answer:

Marlee is correct. They are not congruent because the corresponding legs are not marked as congruent.