Question

do these triangles satisfy hl theorem?
options: clearly they do; obviously not

Explanation:

Step1: Recall HL Theorem

HL (Hypotenuse - Leg) 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. First, we identify the right - triangles (both have a right - angle symbol, so they are right - triangles).

Step2: Analyze the sides

For the first right - triangle: Let's label the legs and hypotenuse. The leg with one tick mark, the leg with two tick marks, and the hypotenuse (the side opposite the right - angle). For the second right - triangle: The leg with one tick mark, the leg with two tick marks, and the hypotenuse. Wait, no—wait, in HL, we need hypotenuse and one leg. Let's check the markings. The first triangle: one leg (horizontal) has 1 tick, another leg (vertical) has 2 ticks, hypotenuse is the slant side. The second triangle: one leg (vertical) has 1 tick, another leg (horizontal) has 2 ticks, hypotenuse is the slant side. Wait, actually, let's see the congruence of hypotenuse and one leg. Wait, no—maybe I misread. Wait, the first triangle: right - angle, one leg (let's say leg \(a\)) with 1 tick, another leg (leg \(b\)) with 2 ticks, hypotenuse \(c\). The second triangle: right - angle, one leg (leg \(a'\)) with 1 tick, another leg (leg \(b'\)) with 2 ticks, hypotenuse \(c'\). Wait, but in HL, we need hypotenuse and one leg. Wait, no—maybe the markings: the first triangle's leg with 1 tick and the second triangle's leg with 1 tick—are they corresponding? The first triangle's leg with 2 ticks and the second triangle's leg with 2 ticks—are they corresponding? Wait, no, the right - triangles: let's check the hypotenuse and one leg. Wait, actually, the two right - triangles: the first has a leg with 1 tick (let's say the horizontal leg) and a leg with 2 ticks (vertical leg), hypotenuse (slant). The second has a leg with 1 tick (vertical leg) and a leg with 2 ticks (horizontal leg), hypotenuse (slant). Wait, but in HL, we need hypotenuse and one leg. Wait, maybe the hypotenuse: are the hypotenuses congruent? And one leg: the leg with 1 tick in the first triangle and the leg with 1 tick in the second triangle—are they congruent? The leg with 2 ticks in the first and the leg with 2 ticks in the second—are they congruent? Wait, no, the problem is: do they satisfy HL? Wait, HL requires hypotenuse and one leg. But here, let's see the markings. The first triangle: right - angle, one leg (let's call it leg 1) with 1 mark, another leg (leg 2) with 2 marks, hypotenuse (H1). The second triangle: right - angle, one leg (leg 1') with 1 mark, another leg (leg 2') with 2 marks, hypotenuse (H2). Wait, but in HL, we need hypotenuse and one leg. Wait, maybe the 1 - marked legs are corresponding legs, and the hypotenuses? Wait, no, maybe I made a mistake. Wait, the two triangles: the first has a horizontal leg with 1 mark, vertical leg with 2 marks, hypotenuse. The second has a vertical leg with 1 mark, horizontal leg with 2 marks, hypotenuse. So the 1 - marked legs: one is horizontal, one is vertical—so they are not corresponding legs? Wait, no, that can't be. Wait, no—maybe the triangles are congruent by HL? Wait, no, HL is hypotenuse and one leg. Wait, maybe the 2 - marked legs are the hypotenuses? No, hypotenuse is the longest side. Wait, no, in a right - triangle, hypotenuse is opposite the right - angle. So in the first triangle, the hypotenuse is the slant side (opposite the right - angle). In the second triangle, the hypotenuse is the slant side (opposite the right - angle). The legs: in the first triangle, the horizontal leg (1 mark) and vertical leg (2 marks). In the second triangle, the vertical leg (1 mark) and horizontal leg (2 marks). So the 1 - marked legs: one is horizontal, one is vertical—so they are not the same type of leg (one is a horizontal leg, one is a vertical leg). The 2 - marked legs: one is vertical, one is horizontal—also not the same type. So the hypotenuse and one leg: do we have hypotenuse congruent and one leg congruent? Wait, no—because the 1 - marked legs are not corresponding (different orientations), and the 2 - marked legs are not corresponding. Wait, but maybe the triangles are congruent by SAS? But the question is about HL. Wait, no—wait, maybe I misread the markings. Wait, the first triangle: right - angle, one leg (let's say the base) with 1 tick, the other leg (height) with 2 ticks, hypotenuse. The second triangle: right - angle, one leg (height) with 1 tick, the other leg (base) with 2 ticks, hypotenuse. So the hypotenuse: is it congruent? And the leg with 1 tick: in the first, it's the base; in the second, it's the height. So they are not the same leg (one is base, one is height). So the HL theorem requires hypotenuse and one leg (either the same leg: either both bases or both heights). But here, the 1 - marked legs are different (base vs height), and the 2 - marked legs are different (height vs base). So they do not satisfy HL. Wait, but wait—maybe the triangles are congruent, but not by HL? Wait, the question is "Do these triangles satisfy HL theorem?". So we need to check if hypotenuse and one leg are congruent. Since the 1 - marked legs are not corresponding (different sides: one is a leg of one type, one is a leg of another type), and the hypotenuse—wait, maybe the hypotenuse is congruent? But we can't tell from the markings. Wait, no, the markings for the legs: the 1 - marked legs are different (one is horizontal, one is vertical), so they are not the same leg. So the HL theorem is not satisfied. So the answer is "obviously not".

Answer:

obviously not