Question

which pair of triangles can be proven congruent by the hl theorem?

Explanation:

Step1: Recall HL Theorem

The Hypotenuse - Leg (HL) Theorem states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two right triangles are congruent. So, we need to identify pairs of right triangles (triangles with a right - angle mark) where the hypotenuse and one leg are marked as congruent.

Step2: Analyze each option

• First, we look for right - angled triangles (triangles with the right - angle symbol). Then we check the markings for hypotenuse and leg congruence.
• The third option (the pair of right - angled triangles with two pairs of congruent legs? Wait, no, let's re - evaluate. Wait, the HL theorem requires hypotenuse and one leg. Let's look at the markings:
• The pair of triangles in the third option (the ones with right angles and two congruent segments on the legs and hypotenuse? Wait, actually, the key is: right triangles, hypotenuse congruent, and one leg congruent.
• Looking at the figures, the pair of right - angled triangles where one leg is marked congruent and the hypotenuse is marked congruent (the third set of triangles, the ones with the right - angle and two congruent marks on the legs and hypotenuse? Wait, no, let's think again. The HL theorem is for right triangles. So first, we need right triangles. Then, hypotenuse and one leg.
• The first pair: two right triangles, one leg marked congruent, but what about hypotenuse? Wait, maybe the third option (the pair of right - angled triangles with two congruent legs and hypotenuse? Wait, no, let's check the markings. The correct pair should be the two right - angled triangles where one leg is congruent (marked with one tick) and the hypotenuse is congruent (marked with two ticks? Wait, no, the HL theorem: in right triangles, hypotenuse (longest side) and one leg. So the pair of right triangles where the hypotenuse is congruent and one leg is congruent. Looking at the options, the third option (the pair of right - angled triangles with the right - angle and the markings for hypotenuse and one leg congruent) is the one that satisfies HL. Wait, actually, the correct pair is the two right - angled triangles where one leg is marked congruent and the hypotenuse is marked congruent. So the pair of right triangles (with the right - angle symbol) that have one leg and hypotenuse congruent. So the answer is the third option (the pair of right - angled triangles with the appropriate markings for HL).

Answer:

The pair of right - angled triangles (the third option in the list of triangle pairs) where the hypotenuse and one leg are congruent (as indicated by the congruence markings) can be proven congruent by the HL theorem. (If we consider the options as A, B, C, D, and the third option is C, then the answer is C. The pair of right - angled triangles with the right - angle symbol and congruent hypotenuse and one leg markings.)