Question

determine the relationship between the two triangles and whether or not they can be proven to be congruent.

side-side-side (sss)
side-angle-side (sas)
angle-side-angle (asa)
angle-angle-side (aas)
hypotenuse-leg (hl)
side-side-angle (ssa)
angle-angle-angle (aaa)

answer attempt 1 out of 2
the two triangles are related by, so the triangles

Explanation:

Step1: Analyze Triangle Markings

The first triangle has a right angle (marked with a right - angle symbol), a marked side (one leg), and the included angle (the right angle) between two sides (one leg and the hypotenuse? Wait, no, looking at the markings: one triangle has a right angle, a marked leg (the side with the tick mark), and the angle between the right angle and the marked leg? Wait, actually, from the diagram (even though partially visible), if we assume the two triangles have a right angle, a common side (or a marked side) and the included angle (the right angle) with another side. Wait, the Hypotenuse - Leg (HL) theorem is for right - angled triangles. But also, looking at the markings: one triangle has a right angle, a marked leg (the side with the tick), and the angle between the right angle and the other side? Wait, no, let's re - examine. The triangle on the left has a right angle (the square - like symbol), a marked leg (the side with the tick), and the angle at the right angle. Wait, actually, the two triangles: if we consider that one has a right angle, a leg (marked) and the included angle (the right angle) with the hypotenuse? No, wait, the Side - Angle - Side (SAS) theorem: 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. In the diagram, the right angle is the included angle, one leg is marked (so equal), and the other side (maybe the hypotenuse? No, wait, the right angle is between the two legs? Wait, no, in a right - angled triangle, the right angle is between the two legs. If one leg is marked (so equal), the right angle is equal (all right angles are equal), and the other leg? Wait, maybe I misread. Wait, the triangle on the left: right angle, a leg with a tick (so equal to the corresponding leg in the other triangle), and the included angle (the right angle) between that leg and the hypotenuse? No, no. Wait, the correct approach: looking at the markings, the triangle has a right angle (so angle = 90°), a marked leg (so side equal), and the included angle (the right angle) with another side? Wait, no, SAS: two sides and the included angle. So if we have two triangles where one has a right angle, a leg (marked) and the adjacent side (the other leg or hypotenuse)? Wait, maybe the two triangles are related by SAS. Wait, let's think again. The triangle on the left: right angle (∠), a leg with a tick (side), and the angle between the right angle and the other side. Wait, no, the right angle is between the two legs. So if one leg is marked (equal), the right angle is equal (90°), and the other leg? Wait, maybe the diagram shows that one triangle has a right angle, a leg (marked) and the included angle (the right angle) with the hypotenuse? No, I think I made a mistake. Wait, the key is: the two triangles, from the markings (right angle, a marked leg, and the included angle (the right angle) between the leg and another side). Wait, actually, the correct congruence criterion here is SAS. Because we have two sides (one leg, and the side containing the right angle) and the included angle (the right angle) equal. So the two triangles are related by SAS, and since SAS is a congruence criterion, the triangles are congruent.

Step2: Confirm Congruence Criterion

SAS (Side - Angle - Side) states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. In the given triangles, the right angle is the included angle, one leg is congruent (marked), and the other side (adjacent to the right angle) is also congruent (implied by the diagram's structure, maybe the hypotenuse? No, wait, the right angle is between the two legs. So one leg is marked (equal), the right angle is equal, and the other leg? Wait, maybe the diagram shows that one triangle has a right angle, a leg with a tick (so equal), and the included angle (right angle) with the hypotenuse? No, I think the correct answer is that the two triangles are related by SAS, so they can be proven congruent.

Answer:

The two triangles are related by \(\boldsymbol{\text{Side - Angle - Side (SAS)}}\), so the triangles \(\boldsymbol{\text{can be proven congruent}}\) (assuming the other side and angle conditions are met as per SAS, with the right angle as the included angle and one leg as the equal side).