Question

determine the relationship between the two triangles and whether or not they can be proven to be congruent. answer the two triangles are related by , so the triangles

Explanation:

Step1: Analyze the triangle

The triangle is a right triangle (has a right angle) and has two sides marked as congruent (the tick marks). For right triangles, the Hypotenuse - Leg (HL) congruence criterion applies when the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle. Also, if we consider the two legs and the right angle, Side - Angle - Side (SAS) could also be a candidate, but since it's a right triangle, HL is more specific for right triangles. However, looking at the marked sides: if we have a right triangle with two legs (the sides forming the right angle) and one of them is congruent, and the hypotenuse? Wait, no, the triangle in the diagram has a right angle and two sides with tick marks. Wait, maybe it's a right triangle with a leg and the hypotenuse? Wait, no, the HL criterion is for hypotenuse and leg. Alternatively, if we have a right triangle, and we have two sides: one leg and the hypotenuse, or two legs. Wait, the triangle shown has a right angle and two sides with congruency marks. Let's think again. The HL (Hypotenuse - Leg) theorem states that if the hypotenuse and one leg of a right triangle are equal to the hypotenuse and one leg of another right triangle, then the triangles are congruent. Alternatively, SAS: if two sides and the included angle (the right angle here) are congruent. But since it's a right triangle, HL is a special case for right triangles. Wait, but let's check the options. The triangle is a right triangle, so HL is applicable. Wait, but maybe the triangle has a leg and the hypotenuse? Or two legs? Wait, the diagram shows a right triangle with two sides marked (the legs? Or one leg and the hypotenuse?). Wait, the right angle is between the two legs. If two legs are marked as congruent, then SAS (since the included angle is the right angle) would be applicable. But also, HL is for hypotenuse and leg. Wait, maybe the triangle has a leg and the hypotenuse. Wait, the problem is about two triangles (even though only one is shown, but the relationship is about two triangles related by a congruence criterion). Assuming that the two triangles are right triangles, and we have the hypotenuse and one leg congruent, then HL. But let's recall the congruence criteria. For right triangles, HL is a valid congruence criterion (SSS, SAS, ASA, AAS, HL are valid; SSA is not, AAA is not). So if the two triangles are right triangles, and we have the hypotenuse and one leg congruent, then HL. Alternatively, if we have two legs and the right angle, SAS. But looking at the diagram, the triangle has a right angle and two sides with tick marks. Let's assume that the two triangles are related by HL. Wait, maybe the answer is HL. Wait, but let's check the options. The options include HL (Hypotenuse - Leg). So the two triangles are related by Hypotenuse - Leg (HL), so the triangles can be proven congruent by HL.

Step2: Confirm the congruence criterion

HL (Hypotenuse - Leg) is a valid congruence criterion for right triangles. If two right triangles have their hypotenuse and one corresponding leg congruent, then they are congruent. Since the triangle in the diagram is a right triangle (has a right angle) and we are comparing two such triangles, the relationship is HL, and they can be proven congruent by HL.

Answer:

The two triangles are related by Hypotenuse - Leg (HL), so the triangles can be proven congruent by HL. (So the first blank is "Hypotenuse - Leg (HL)")