4-Right Triangle Congruence.pdf - Kuta Software - Free Printable

Let's go through each of the 10 problems on this Right Triangle Congruence worksheet and determine whether the two triangles in each pair are congruent. We'll use the right triangle congruence theorems, which include:

- HL (Hypotenuse-Leg): If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent.
- LL (Leg-Leg): If the two legs of one right triangle are congruent to the two legs of another, then the triangles are congruent.
- HA (Hypotenuse-Angle): If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and an acute angle of another, the triangles are congruent.
- LA (Leg-Angle): If one leg and an acute angle of one right triangle are congruent to one leg and an acute angle of another, the triangles are congruent.

We’ll analyze each pair based on markings (sides and angles marked with tick marks or symbols).

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1)

Two right triangles. One has a leg and hypotenuse marked with tick marks. The other has corresponding sides marked similarly.

- Both have right angles.
- One leg and the hypotenuse are marked as congruent.
- This matches HL (Hypotenuse-Leg) congruence.

✔ Congruent by HL

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2)

Two right triangles. One has a leg and the hypotenuse marked; the other has a leg and hypotenuse marked — but not necessarily matching.

Wait: Let’s look carefully.

Left triangle: Right angle, one leg marked with one tick, hypotenuse marked with two ticks.

Right triangle: Right angle, one leg marked with one tick, hypotenuse marked with two ticks.

So:
- One leg is congruent (one tick).
- Hypotenuse is congruent (two ticks).
- Right angles are both present.

✔ Congruent by HL

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3)

Two right triangles. Each has a leg marked with one tick, and the hypotenuse marked with two ticks.

But wait — they’re drawn side-by-side. Are they mirror images?

Yes — both have:
- Right angles
- One leg marked with one tick
- Hypotenuse marked with two ticks

So again, HL applies.

✔ Congruent by HL

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4)

Two right triangles connected at a point. They share a common vertex, but are rotated.

One triangle has:
- Right angle
- One leg marked with one tick
- Hypotenuse marked with two ticks

Other triangle:
- Right angle
- Same leg and hypotenuse marked identically

So:
- One leg ≅ one leg
- Hypotenuse ≅ hypotenuse
- Right angles → so HL

✔ Congruent by HL

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5)

Two right triangles forming a rectangle? Wait — it looks like a square split by a diagonal.

Actually, it's a single figure: a rectangle (or square) with a diagonal drawn.

The two triangles are:
- Both right triangles
- Share the diagonal (hypotenuse)
- Two legs are sides of the rectangle

But are the two triangles congruent?

Yes — because:
- Both have right angles
- Two legs are equal (opposite sides of rectangle), and if it's a square, all sides equal — but even if rectangle, opposite sides are equal.

Wait: In a rectangle, opposite sides are equal. So:
- Leg 1 = Leg 1 (shared)
- Leg 2 = Leg 2 (opposite side)

So both triangles have:
- Two legs congruent → LL congruence

✔ Congruent by LL

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6)

Two right triangles, joined at a point. One triangle has:
- Right angle
- One leg marked with one tick
- Hypotenuse marked with two ticks

Other triangle:
- Right angle
- One leg marked with one tick
- Hypotenuse marked with two ticks

Same markings → HL

✔ Congruent by HL

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7)

Two separate right triangles.

Each has:
- A right angle
- One leg marked with "100" (so same length)
- Another leg marked with "100"? Wait — no.

Wait: First triangle: one leg marked 100, and the other leg unmarked.

Second triangle: one leg marked 100, and the other leg unmarked.

But both have:
- Right angles
- One leg = 100
- But we don’t know about the other leg or hypotenuse.

Are any other sides marked? No.

Only one leg is given as 100, and no other information.

So unless we assume something else, not enough info.

But wait — are there tick marks?

Looking closely: the first triangle has one leg labeled "100", the second has one leg labeled "100". But no tick marks indicating other sides.

But also: are the angles marked? No.

So only one leg is known to be equal. We don't know anything about the other leg or hypotenuse.

✘ Not enough information → Not congruent (cannot conclude)

Wait — but maybe the triangles are drawn such that the other sides are implied?

No — without markings, we can’t assume.

So unless more is given, no.

But wait — perhaps the triangles are positioned so that the hypotenuses are shared or something? No, they're separate.

So only one leg is equal. Not sufficient.

✘ Not congruent (only one side known to be equal)

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Wait — actually, let's recheck: both triangles have a leg of 100, and both have right angles, but we don't know if the other leg or hypotenuse is equal.

So no — cannot conclude congruence.

✘ Not congruent

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8)

Two right triangles connected at a vertex.

First triangle: right angle, one leg marked with one tick.

Second triangle: right angle, one leg marked with one tick, and another side marked with two ticks.

Wait — let's see:

- Both have right angles
- One leg marked with one tick in each → so one leg congruent
- But second triangle has an additional marking: a side with two ticks, but first triangle doesn't have that marking

So unless the two-tick side corresponds to something, we can't say.

But the way it's drawn: the two triangles share a side? Or are adjacent?

It looks like they are joined at a point, but not overlapping.

Wait — actually, the figure shows two right triangles sharing a leg?

No — one triangle has a leg marked with one tick, and the other has a leg marked with one tick, and another side marked with two ticks.

But no indication that the two-tick side is the same in both.

Also, the angles are not marked.

So only one leg is congruent, and right angles — but that's not enough.

Wait — could it be LA?

If one leg and an acute angle are congruent?

But no angles are marked.

So no.

Only one side marked (the leg), and right angles.

But unless the other leg or hypotenuse is known, can't prove congruence.

✘ Not congruent

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Wait — actually, look again: in problem 8, the two triangles are drawn sharing a common side? No — they appear to be adjacent.

But the first triangle has a leg with one tick, and the second triangle has a leg with one tick — so those legs are congruent.

But nothing else is marked.

So only one leg and the right angle — not enough.

✘ Not congruent

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9)

Two right triangles, overlapping? Or intersecting?

They cross each other, but both have:
- Right angles
- One leg marked with one tick
- Hypotenuse marked with two ticks

So:
- One leg ≅ one leg (tick mark)
- Hypotenuse ≅ hypotenuse (two ticks)
- Right angles

So HL applies.

✔ Congruent by HL

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10)

Two right triangles, joined at a point.

Both have:
- Right angles
- One leg marked with one tick
- Hypotenuse marked with two ticks

So again:
- One leg ≅ one leg
- Hypotenuse ≅ hypotenuse
- Right angles → HL

✔ Congruent by HL

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✔ Final Answers:

1) Congruent by HL
2) Congruent by HL
3) Congruent by HL
4) Congruent by HL
5) Congruent by LL
6) Congruent by HL
7) Not congruent (only one leg equal, insufficient)
8) Not congruent (only one leg equal, insufficient)
9) Congruent by HL
10) Congruent by HL

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🔍 Explanation Summary:

- For HL: Need hypotenuse and one leg marked as congruent.
- For LL: Need both legs marked.
- For HA or LA: Need angle and side, but angles must be marked.
- In problems 7 and 8, only one leg is marked as equal (or same measure), but no other sides or angles are confirmed, so insufficient to prove congruence.

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