Question

which criterion, if any, could prove the triangles are congruent?
a. asa
b. sas
c. hl
d. none

Explanation:

The triangles are right - angled. We know that one leg (the marked side) is equal, and the right angle is common. For right - angled triangles, the Hypotenuse - Leg (HL) criterion states that if the hypotenuse and one leg of a right - triangle are equal to the hypotenuse and one leg of another right - triangle, the triangles are congruent. Here, we can consider the right angle, the equal leg, and we can infer about the hypotenuse or use HL. Wait, actually, let's re - examine. The two triangles are right - angled, they have one leg equal (the marked segments), and the right angle. But also, for HL, we need hypotenuse and leg. Wait, no, actually, in right - angled triangles, if we have a leg and the hypotenuse? Wait, no, looking at the diagram, the two right - angled triangles have a pair of equal legs (the marked ones) and the right angle. But also, the hypotenuse? Wait, no, maybe I made a mistake. Wait, the HL (Hypotenuse - Leg) theorem: In two right - angled triangles, if the hypotenuse and one leg are equal, then the triangles are congruent. But here, we have a right angle, one leg equal, and what about the hypotenuse? Wait, no, maybe the triangles have the hypotenuse? Wait, no, let's think again. The two triangles are right - angled, so $\angle C=\angle F = 90^{\circ}$, the leg $BC = EF$ (marked), and we can consider the hypotenuse? Wait, no, maybe the answer is HL. Wait, the HL criterion applies to right - angled triangles. So the two right - angled triangles, with one leg equal and the hypotenuse? Wait, no, maybe the marked side is a leg, and the hypotenuse? Wait, no, let's check the options. The options are ASA, SAS, HL, or none. ASA requires two angles and the included side. SAS requires two sides and the included angle. HL is for right - angled triangles: hypotenuse and leg. In the diagram, we have two right - angled triangles, one leg is equal (the marked segment), and the right angle. But also, the hypotenuse? Wait, no, maybe the triangles have the hypotenuse equal? Wait, no, the diagram shows two right - angled triangles with a common - length leg (marked) and right angles. So for right - angled triangles, HL (Hypotenuse - Leg) can be used if we have hypotenuse and leg. Wait, maybe the other side is the hypotenuse? Wait, no, let's recall: HL states that in $Rt\triangle ABC$ and $Rt\triangle DEF$, if $AC = DF$ (hypotenuse) and $BC = EF$ (leg), then $\triangle ABC\cong\triangle DEF$. So in the given diagram, the two right - angled triangles have a leg equal (marked) and the right angle. But also, the hypotenuse? Wait, maybe the triangles are congruent by HL. So the correct option is C.

Answer:

C. HL