Worksheet demonstrating the Angle-Angle-Side (AAS) congruence theorem with sample proofs and diagrams.

Let's solve the Angle-Angle-Side (AAS) problem step by step, based on the image you've uploaded.

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🔷 Part 1: Understanding AAS Congruence

The Angle-Angle-Side (AAS) Congruence Theorem states:

> If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the triangles are congruent.

In the diagram:
- Triangle $ \triangle ABC $ and $ \triangle DEF $
- There are right angles at $ B $ and $ E $ (indicated by the small square)
- Angles at $ A $ and $ D $ are marked as equal
- Sides $ BC $ and $ EF $ are marked as equal (the non-included sides)

So we fill in the blanks:

> If
> $ \angle A \cong \angle D $ (Angle)
> $ \angle B \cong \angle E $ (Angle)
> $ BC \cong EF $ (Side)
> then, $ \triangle ABC \cong \triangle DEF $

✔ Note: Since $ BC $ is opposite $ \angle A $, and $ EF $ is opposite $ \angle D $, they are non-included sides — this satisfies AAS.

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🔷 Part 2: Sample AAS Proof

We are given:

> Given: $ \overline{YZ} $ bisects $ \angle WYX $, and $ \angle YWZ \cong \angle YXZ $
> Prove: $ \triangle WYZ \cong \triangle XYZ $

Let’s analyze the diagram and write the proof using AAS.

#### Step-by-step Proof

| Statements | Reasons |
|----------|--------|
| 1. $ \overline{YZ} $ bisects $ \angle WYX $ | 1. Given |
| 2. $ \angle WYZ \cong \angle XYZ $ | 2. Definition of angle bisector |
| 3. $ \angle YWZ \cong \angle YXZ $ | 3. Given |
| 4. $ \overline{YZ} \cong \overline{YZ} $ | 4. Reflexive Property of Congruence |
| 5. $ \triangle WYZ \cong \triangle XYZ $ | 5. AAS Congruence Theorem |

Wait! Let's double-check: Is this AAS?

We have:
- $ \angle WYZ \cong \angle XYZ $ (from bisector)
- $ \angle YWZ \cong \angle YXZ $ (given)
- $ \overline{YZ} $ is common to both triangles

But note: $ \overline{YZ} $ is opposite $ \angle W $ and $ \angle X $, so it is not between the two angles we're using.

So yes — we have:
- Two angles: $ \angle WYZ \cong \angle XYZ $
- Another angle: $ \angle YWZ \cong \angle YXZ $
- And a non-included side $ \overline{YZ} $ that is shared

Thus, AAS applies, because:
- Two angles and a non-included side (the side not between the two angles) are congruent.

✔ So the proof is valid.

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✔ Final Answer for the Blank:

> If
> $ \angle A \cong \angle D $ (Angle)
> $ \angle B \cong \angle E $ (Angle)
> $ BC \cong EF $ (Side)
> then, $ \triangle ABC \cong \triangle DEF $

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✔ Completed Proof Table:

| Statements | Reasons |
|----------|--------|
| 1. $ \overline{YZ} $ bisects $ \angle WYX $ | 1. Given |
| 2. $ \angle WYZ \cong \angle XYZ $ | 2. Definition of angle bisector |
| 3. $ \angle YWZ \cong \angle YXZ $ | 3. Given |
| 4. $ \overline{YZ} \cong \overline{YZ} $ | 4. Reflexive Property |
| 5. $ \triangle WYZ \cong \triangle XYZ $ | 5. AAS Congruence Theorem |

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📌 Key Takeaway:

AAS works when:
- Two angles are congruent
- One non-included side (i.e., a side not between the two angles) is congruent
- Then the triangles are congruent.

This is equivalent to ASA (Angle-Side-Angle), but with the side not between the angles.

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