Practice worksheet for proving triangle congruence using ASA, AAS, SAS, and SSS criteria.

Problem Overview:

The task involves identifying the correct congruence criteria for pairs of triangles in each problem. The congruence criteria to be used are:

- SSS (Side-Side-Side): All three sides of one triangle are equal to the corresponding sides of another triangle.
- SAS (Side-Angle-Side): Two sides and the included angle of one triangle are equal to the corresponding parts of another triangle.
- ASA (Angle-Side-Angle): Two angles and the included side of one triangle are equal to the corresponding parts of another triangle.
- AAS (Angle-Angle-Side): Two angles and a non-included side of one triangle are equal to the corresponding parts of another triangle.

We will analyze each pair of triangles step by step and determine the appropriate congruence criterion.

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Solution:

#### Problem 1:
- Triangles: \( \triangle ABC \) and \( \triangle DEF \)
- Given: \( AB = DE \), \( BC = EF \), \( AC = DF \)
- Reasoning: All three sides of \( \triangle ABC \) are equal to the corresponding sides of \( \triangle DEF \).
- Conclusion: \( \triangle ABC \cong \triangle DEF \) by SSS.

#### Problem 2:
- Triangles: \( \triangle ABC \) and \( \triangle DEF \)
- Given: \( AB = DE \), \( \angle B = \angle E \), \( BC = EF \)
- Reasoning: Two sides and the included angle of \( \triangle ABC \) are equal to the corresponding parts of \( \triangle DEF \).
- Conclusion: \( \triangle ABC \cong \triangle DEF \) by SAS.

#### Problem 3:
- Triangles: \( \triangle ABC \) and \( \triangle DEF \)
- Given: \( \angle A = \angle D \), \( AB = DE \), \( \angle B = \angle E \)
- Reasoning: Two angles and the included side of \( \triangle ABC \) are equal to the corresponding parts of \( \triangle DEF \).
- Conclusion: \( \triangle ABC \cong \triangle DEF \) by ASA.

#### Problem 4:
- Triangles: \( \triangle ABC \) and \( \triangle DEF \)
- Given: \( \angle A = \angle D \), \( \angle C = \angle F \), \( BC = EF \)
- Reasoning: Two angles and a non-included side of \( \triangle ABC \) are equal to the corresponding parts of \( \triangle DEF \).
- Conclusion: \( \triangle ABC \cong \triangle DEF \) by AAS.

#### Problem 5:
- Triangles: \( \triangle ABC \) and \( \triangle DEF \)
- Given: \( AB = DE \), \( BC = EF \), \( \angle B = \angle E \)
- Reasoning: Two sides and the included angle of \( \triangle ABC \) are equal to the corresponding parts of \( \triangle DEF \).
- Conclusion: \( \triangle ABC \cong \triangle DEF \) by SAS.

#### Problem 6:
- Triangles: \( \triangle ABC \) and \( \triangle DEF \)
- Given: \( \angle A = \angle D \), \( \angle B = \angle E \), \( BC = EF \)
- Reasoning: Two angles and a non-included side of \( \triangle ABC \) are equal to the corresponding parts of \( \triangle DEF \).
- Conclusion: \( \triangle ABC \cong \triangle DEF \) by AAS.

#### Problem 7:
- Triangles: \( \triangle ABC \) and \( \triangle DEF \)
- Given: \( AB = DE \), \( \angle B = \angle E \), \( \angle C = \angle F \)
- Reasoning: Two angles and a non-included side of \( \triangle ABC \) are equal to the corresponding parts of \( \triangle DEF \).
- Conclusion: \( \triangle ABC \cong \triangle DEF \) by AAS.

#### Problem 8:
- Triangles: \( \triangle ABC \) and \( \triangle DEF \)
- Given: \( \angle A = \angle D \), \( AB = DE \), \( \angle C = \angle F \)
- Reasoning: Two angles and a non-included side of \( \triangle ABC \) are equal to the corresponding parts of \( \triangle DEF \).
- Conclusion: \( \triangle ABC \cong \triangle DEF \) by AAS.

#### Problem 9:
- Triangles: \( \triangle ABC \) and \( \triangle DEF \)
- Given: \( AB = DE \), \( BC = EF \), \( AC = DF \)
- Reasoning: All three sides of \( \triangle ABC \) are equal to the corresponding sides of \( \triangle DEF \).
- Conclusion: \( \triangle ABC \cong \triangle DEF \) by SSS.

#### Problem 10:
- Triangles: \( \triangle ABC \) and \( \triangle DEF \)
- Given: \( \angle A = \angle D \), \( \angle B = \angle E \), \( AB = DE \)
- Reasoning: Two angles and the included side of \( \triangle ABC \) are equal to the corresponding parts of \( \triangle DEF \).
- Conclusion: \( \triangle ABC \cong \triangle DEF \) by ASA.

#### Problem 11:
- Triangles: \( \triangle ABC \) and \( \triangle DEF \)
- Given: \( AB = DE \), \( \angle B = \angle E \), \( BC = EF \)
- Reasoning: Two sides and the included angle of \( \triangle ABC \) are equal to the corresponding parts of \( \triangle DEF \).
- Conclusion: \( \triangle ABC \cong \triangle DEF \) by SAS.

#### Problem 12:
- Triangles: \( \triangle ABC \) and \( \triangle DEF \)
- Given: \( \angle A = \angle D \), \( AB = DE \), \( \angle B = \angle E \)
- Reasoning: Two angles and the included side of \( \triangle ABC \) are equal to the corresponding parts of \( \triangle DEF \).
- Conclusion: \( \triangle ABC \cong \triangle DEF \) by ASA.

#### Problem 13:
- Triangles: \( \triangle ABC \) and \( \triangle DEF \)
- Given: \( AB = DE \), \( BC = EF \), \( AC = DF \)
- Reasoning: All three sides of \( \triangle ABC \) are equal to the corresponding sides of \( \triangle DEF \).
- Conclusion: \( \triangle ABC \cong \triangle DEF \) by SSS.

#### Problem 14:
- Triangles: \( \triangle ABC \) and \( \triangle DEF \)
- Given: \( \angle A = \angle D \), \( \angle B = \angle E \), \( BC = EF \)
- Reasoning: Two angles and a non-included side of \( \triangle ABC \) are equal to the corresponding parts of \( \triangle DEF \).
- Conclusion: \( \triangle ABC \cong \triangle DEF \) by AAS.

#### Problem 15:
- Triangles: \( \triangle ABC \) and \( \triangle DEF \)
- Given: \( AB = DE \), \( \angle B = \angle E \), \( BC = EF \)
- Reasoning: Two sides and the included angle of \( \triangle ABC \) are equal to the corresponding parts of \( \triangle DEF \).
- Conclusion: \( \triangle ABC \cong \triangle DEF \) by SAS.

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Final Answer:

\[
\boxed{
\begin{array}{c|c}
\text{Problem} & \text{Congruence Criterion} \\
\hline
1 & SSS \\
2 & SAS \\
3 & ASA \\
4 & AAS \\
5 & SAS \\
6 & AAS \\
7 & AAS \\
8 & AAS \\
9 & SSS \\
10 & ASA \\
11 & SAS \\
12 & ASA \\
13 & SSS \\
14 & AAS \\
15 & SAS \\
\end{array}
}
\]