Hypotenuse-Leg Congruence Theorem: Determine if the given pairs of right triangles are congruent using the HL theorem.

Let's analyze each pair of triangles shown in the image and determine whether they are congruent using the Hypotenuse-Leg (HL) Congruence Theorem or other applicable congruence theorems.

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🔷 Hypotenuse-Leg (HL) Congruence Theorem:

> 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 triangles are congruent.

This theorem only applies to right triangles.

We'll go through each numbered pair:

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1)

- Two right triangles sharing a common side.
- One leg is marked equal (red tick), and both have right angles.
- The hypotenuses are not marked as equal.
- Only one leg is marked as equal.
- The shared side might be a leg, but it’s unclear if it's the same leg in both.

Wait — actually, this looks like two right triangles that share a common leg and have a right angle. But:
- One leg is marked with a red tick (same length).
- The other leg is not marked.
- Hypotenuses are not marked.

But look closely: the shared segment appears to be a leg of both triangles, and the red tick marks indicate one leg is equal.

However, we don't know if the hypotenuses are equal. So we cannot use HL unless both the hypotenuse and one leg are known to be equal.

➡️ Not enough information to apply HL.

Also, no other sides or angles are marked.

✔ Conclusion: Not necessarily congruent — insufficient information.

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2)

- Two right triangles.
- Right angles are marked.
- One leg is marked equal (red tick).
- Hypotenuses are not marked.
- The other leg is not marked.

So only one leg is marked as equal, and nothing about the hypotenuses.

➡️ Cannot use HL.

Also, no other sides or angles match.

✔ Conclusion: Not congruent — only one leg marked.

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6)

- Two right triangles.
- Both have right angles marked.
- One leg is marked with double red ticks, meaning it's equal in both.
- The hypotenuse is also marked with double red ticks (same length).

So:
- Right angles → right triangles
- Hypotenuse ≅ hypotenuse
- One leg ≅ one leg

➡️ This satisfies the Hypotenuse-Leg (HL) Congruence Theorem

✔ Conclusion: Yes, congruent by HL.

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8)

- A quadrilateral split into two right triangles.
- Both triangles have right angles.
- One leg is marked with a single red tick.
- The other leg is marked with a double red tick (same in both).
- The hypotenuses are not marked.

But wait — the figure shows a parallelogram-like shape with diagonals?

Actually, it's a quadrilateral with one diagonal drawn, forming two right triangles.

- Each triangle has a right angle.
- One leg is marked equal (single tick).
- The other leg is marked equal (double tick).
- So both legs are marked equal.

Since both legs are equal and both are right triangles, we can use Leg-Leg (LL) — which is a special case of SAS for right triangles.

But note: HL requires hypotenuse and one leg, but here we have both legs.

So this is SAS (since the included angle is the right angle), or LL (a corollary of SAS for right triangles).

But HL does not apply here because we don’t know the hypotenuse is equal.

However, since both legs are congruent and the included angle (right angle) is congruent, we can say:

➡️ Congruent by SAS (or LL)

But the question asks us to justify with a congruence theorem — and HL is mentioned, but this is not HL.

✔ Conclusion: Yes, congruent, but by SAS (not HL). Since the problem focuses on HL, maybe this is not what they want.

But let's see: Are the hypotenuses equal? Not marked. But since both legs are equal, by Pythagoras, hypotenuses must be equal.

So technically, we could use HL if we infer hypotenuse equality.

But since the hypotenuse is not marked, and we're supposed to go by markings, we cannot assume it.

So better to say: Yes, congruent by SAS (since two legs and included right angle are equal).

But again, HL is not the correct justification here.

So if the task is specifically about HL, then this may not qualify.

But the question says: “justify your response with a congruence theorem” — so any valid one.

✔ Conclusion: Yes, congruent by SAS (or LL).

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10)

- Two right triangles.
- Right angles marked.
- One leg marked with a single red tick.
- Another leg marked with a double red tick.
- Hypotenuses are not marked.

Wait — the hypotenuse of one triangle is marked with a single tick, and the other triangle's leg is marked with a single tick.

Wait — let’s examine carefully.

In triangle 10:
- Left triangle: right angle at bottom.
- One leg is marked with two red ticks.
- The other leg is marked with one red tick.
- The hypotenuse is not marked.

Right triangle:
- Right angle at top.
- One leg marked with one red tick.
- Other leg marked with two red ticks.

So:
- One leg of left = one leg of right (1 tick)
- Other leg of left = other leg of right (2 ticks)
- Right angles are equal

So again, both legs are equal, and included angle is right angle → SAS or LL

But hypotenuses are not marked.

So again, not HL.

✔ Conclusion: Yes, congruent by SAS (LL)

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✔ Summary:

| Pair | Congruent? | Justification |
|------|------------|-------------|
| 1) | ✘ No | Only one leg marked; hypotenuse not marked; insufficient info |
| 2) | ✘ No | Only one leg marked; hypotenuse not marked |
| 6) | ✔ Yes | HL: hypotenuse and one leg marked equal, right triangles |
| 8) | ✔ Yes | SAS (two legs and included right angle) |
| 10) | ✔ Yes | SAS (two legs and included right angle) |

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🟩 Final Answer:

Only Pair 6 is congruent by the Hypotenuse-Leg (HL) Congruence Theorem.

Other pairs are congruent by SAS (or LL) but not by HL, because the hypotenuse is not marked as equal.

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✔ Therefore:

- Pair 6: Yes, congruent by HL.
- All others: Not necessarily congruent by HL, though some are congruent by other theorems.

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If the question is asking specifically about HL, then only #6 qualifies.

But if it's asking generally, then #8 and #10 are also congruent, just not by HL.

Given the title "Hypotenuse-Leg Congruence Theorem", likely the focus is on identifying which pairs satisfy HL.

So final boxed answer:

> ✔ Only pair 6 is congruent by the Hypotenuse-Leg (HL) Congruence Theorem.
> - It has a right angle, hypotenuse marked equal, and one leg marked equal.
> - All other pairs lack either the hypotenuse or sufficient markings for HL.