Question

here are two right triangles. one pair of corresponding sides is marked congruent. which statement is true? you would be able to conclude that the triangles were congruent by the hl congruence theorem if you also knew that rs≅vu. you already have enough information to conclude that the triangles are congruent by the asa congruence theorem. you would be able to conclude that the triangles were congruent by the hl congruence theorem if you also knew that rt≅vt. you already have enough information to conclude that the triangles are congruent by the hl congruence theorem.

Explanation:

Step1: Recall HL congruence theorem

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.

Step2: Analyze each option

• Option 1: If \(RS\cong VU\), we have two right - triangles \(\triangle RBT\) and \(\triangle VUT\). We know \(BT = TU\) (marked congruent sides) and \(\angle B=\angle U = 90^{\circ}\). But \(RS\) and \(VU\) are not hypotenuses or legs in the context of the two right - triangles \(\triangle RBT\) and \(\triangle VUT\).
• Option 2: ASA (Angle - Side - Angle) congruence theorem requires two angles and the included side. We only know one pair of right angles (\(\angle B=\angle U = 90^{\circ}\)) and one pair of sides (\(BT = TU\)). We don't have enough angles for ASA.
• Option 3: If \(RT\cong VT\), in right - triangles \(\triangle RBT\) and \(\triangle VUT\), we have \(\angle B=\angle U = 90^{\circ}\), \(BT = TU\) (given) and \(RT\cong VT\) (hypotenuses). By the HL congruence theorem (\(HL\): hypotenuse (\(RT\) and \(VT\)) and leg (\(BT\) and \(TU\))), \(\triangle RBT\cong\triangle VUT\).
• Option 4: We only know one pair of sides (\(BT = TU\)) and one pair of right angles. We need either another pair of sides (one hypotenuse and one leg for \(HL\)) or two pairs of angles and one pair of sides (for \(ASA\) etc.).

Answer:

You would be able to conclude that the triangles were congruent by the HL congruence theorem if you also knew that \(RT\cong VT\).