Printable worksheet for identifying congruent right triangles using geometric postulates.

Let's solve each problem step by step. The goal is to identify the correct postulate (or theorem) that proves each pair of right triangles congruent.

We are dealing with right triangles, so we can use special congruence theorems for right triangles:

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

But in most cases, especially when two sides or angles are marked, HL, LL, or LA are commonly used.

Let’s analyze each diagram:

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1)

Two right triangles sharing a common side (the vertical line), with:
- Right angles at the bottom corners.
- One leg marked equal (the vertical segment).
- The hypotenuses appear to be shared? Wait — actually, they’re two separate right triangles sharing a common vertex and one leg.

Wait — looking carefully:
Each triangle has:
- A right angle at the base.
- A marked leg (vertical) that is equal.
- A hypotenuse that appears to be shared? No — they're separate.

Actually, the two triangles share a common side (the vertical leg), and both have:
- A right angle.
- One leg marked as equal.
- The other leg seems not marked.

Wait — let's look again:
There is a common vertical leg, and one other leg is marked with a tick mark — but it's only on one triangle?

No — wait: both triangles have a leg marked with a single tick, and the other leg is unmarked. But the shared leg is not marked.

Actually, upon closer inspection:
- Both triangles have a right angle.
- One leg (the vertical one) is shared — so it's congruent by reflexive property.
- The other leg (horizontal) is marked with one tick in both triangles → so those legs are congruent.

So both legs are congruent → this is LL (Leg-Leg).

✔ Answer: LL (Leg-Leg)

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2)

Two right triangles:
- One triangle has a right angle at the top-left corner.
- The other has a right angle at the bottom-right corner.
- There is a perpendicular from a point to a line, forming two right triangles.
- Markings:
- A right angle is shown in both.
- A segment from the top to the base is drawn, and there’s a tick mark on the vertical leg of the left triangle and the horizontal leg of the right triangle?
- Actually, there’s a common segment (from the top to the intersection point), and it's marked with a tick in both triangles → so this segment is shared.

Wait — better:
The figure shows a triangle split into two right triangles by an altitude from the right angle to the hypotenuse.

So:
- Both are right triangles.
- They share the altitude (which is a leg in both).
- The hypotenuse of each is part of the original triangle’s legs.

But markings:
- The altitude is marked with a tick in both → so it's congruent.
- Also, the angle at the vertex where the altitude meets the hypotenuse is a right angle.
- Additionally, the original hypotenuse is split into two segments — but only one segment is marked with a tick.

Wait — actually, both segments of the hypotenuse are marked with one tick — meaning they are equal.

So:
- Each triangle has:
- A right angle.
- One leg = the altitude (common, so equal).
- Another leg = the segment of the hypotenuse — and both segments are marked equal.

So both legs are congruent → LL.

✔ Answer: LL (Leg-Leg)

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3)

Two right triangles formed by a vertical line splitting an isosceles triangle.

- Both triangles have a right angle at the base.
- The base is split into two equal parts (since it's symmetric), so the legs along the base are equal.
- The vertical segment is shared → common leg.
- The hypotenuses are the slanted sides — marked with ticks → equal.

So:
- Both triangles have:
- Right angles.
- One leg = vertical (common).
- Hypotenuse = marked equal.

So: HL (Hypotenuse-Leg)

✔ Answer: HL (Hypotenuse-Leg)

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4)

Two right triangles connected at a point, forming a "butterfly" shape.

- Both are right triangles.
- The right angles are marked.
- One leg in each is marked with a tick → so those legs are equal.
- The hypotenuses are also marked with a tick → equal.

So:
- Hypotenuse and one leg are congruent → HL

✔ Answer: HL (Hypotenuse-Leg)

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5)

Two right triangles overlapping.

- Right angles are marked.
- One leg in each triangle is marked with a tick → equal.
- The hypotenuses are marked with a tick → equal.

So again: HL

✔ Answer: HL (Hypotenuse-Leg)

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6)

An isosceles triangle with altitude from apex to base.

- Two right triangles formed.
- Shared altitude (vertical leg) → common.
- Base is split into two equal parts → marked equal.
- So both legs are equal → LL

✔ Answer: LL (Leg-Leg)

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7)

A rectangle with diagonal drawn.

- Two right triangles formed by diagonal.
- Both have:
- Right angles at corners.
- Opposite sides of rectangle are equal → so legs are equal.
- Diagonal is shared → hypotenuse.

So:
- Both legs equal → LL
- Or: hypotenuse and one leg? But both legs are equal.

Since both legs are congruent, LL applies.

Alternatively, since all three sides are equal (rectangle sides and diagonal), SSS could work, but we’re using right triangle theorems.

✔ Answer: LL (Leg-Leg)

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8)

Two right triangles sharing a common vertex.

- Right angles are marked.
- One leg is marked with a tick in both → equal.
- An acute angle is marked in both → equal.

So:
- One leg and an acute angle → LA (Leg-Acute angle)

Note: Since both are right triangles, and we have:
- One leg
- One acute angle (adjacent or opposite?)

But if one leg and an acute angle are congruent, then LA applies.

✔ Answer: LA (Leg-Acute angle)

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✔ Final Answers:

1) LL
2) LL
3) HL
4) HL
5) HL
6) LL
7) LL
8) LA

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Summary:

| Problem | Postulate |
|--------|----------|
| 1 | LL |
| 2 | LL |
| 3 | HL |
| 4 | HL |
| 5 | HL |
| 6 | LL |
| 7 | LL |
| 8 | LA |

Let me know if you'd like a visual explanation for any!