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 problem based on markings (tick marks for congruent sides, angles, and right angles).

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1)

- Two right triangles.
- One leg is marked with a tick mark in both triangles.
- The hypotenuse is also marked as equal in both.
- Right angles are shown.

✔ Congruent by HL (Hypotenuse-Leg)
→ Hypotenuse and one leg are congruent.

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2)

- Two right triangles.
- One leg is marked equal (one tick).
- The other leg is not marked.
- But one acute angle is marked equal (both have same angle symbol).

Wait: Only one leg and one acute angle are marked? Let’s look closely:
- One leg is marked equal (tick).
- One acute angle is marked equal.
- Both have right angles.

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

✔ Congruent by LA

> Note: Since it's a right triangle, knowing one leg and one acute angle determines the rest.

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3)

- Two right triangles.
- Both legs are marked with tick marks (same number of ticks).
- Right angles shown.

✔ Congruent by LL (Leg-Leg)
→ Both legs are congruent.

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4)

- Two right triangles, but they are positioned differently.
- One leg is marked with a tick in both.
- The hypotenuse is also marked with a tick.
- Right angles are shown.

✔ Congruent by HL
→ Hypotenuse and one leg match.

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5)

- Two right triangles inside a rectangle.
- Diagonal splits the rectangle into two right triangles.
- Both share the diagonal (hypotenuse), and the legs are sides of the rectangle (equal opposite sides).

→ So both legs are equal (rectangle properties), and right angles are there.

✔ Congruent by LL
→ Both legs are equal.

Alternatively, since both hypotenuses are the same diagonal, and legs are equal, HL or LL works.

But clearly: LL is simplest.

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6)

- Two right triangles sharing a common side.
- One leg is marked with a tick in both.
- The hypotenuse is also marked with a tick.
- Right angles shown.

✔ Congruent by HL
→ Hypotenuse and one leg are congruent.

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7)

- Two right triangles.
- One leg is marked with "100" in both.
- Another leg is marked with "80" in both.
- Right angles shown.

→ Both legs are equal → LL

✔ Congruent by LL

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8)

- Two right triangles.
- One leg is marked with a tick in both.
- The hypotenuse is marked with a tick in both.
- Right angles shown.

✔ Congruent by HL

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9)

- Two right triangles overlapping.
- One leg is marked with a tick in both.
- The hypotenuse is marked with a tick in both.
- Right angles shown.

✔ Congruent by HL

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10)

- Two right triangles.
- One leg is marked with a tick in both.
- The other leg is also marked with a tick in both.
- Right angles shown.

→ Both legs are marked as equal.

✔ Congruent by LL

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✔ Final Answers:

| Problem | Congruent? | Reason |
|--------|------------|--------|
| 1) | Yes | HL (Hypotenuse-Leg) |
| 2) | Yes | LA (Leg-Angle) |
| 3) | Yes | LL (Leg-Leg) |
| 4) | Yes | HL |
| 5) | Yes | LL |
| 6) | Yes | HL |
| 7) | Yes | LL |
| 8) | Yes | HL |
| 9) | Yes | HL |
| 10) | Yes | LL |

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🔍 Summary of Congruence Reasons Used:

- HL: Problems 1, 4, 6, 8, 9
- LL: Problems 3, 5, 7, 10
- LA: Problem 2

> All pairs are congruent based on the given markings.

Let me know if you'd like a visual explanation or need help drawing the reasoning!