Congruent Triangle Proofs Practice Worksheet with three problems involving triangle congruence and geometric reasoning.

Problem 1: Prove that \( \angle J \cong \angle H \)

#### Given:
- \( \overline{JG} \cong \overline{HI} \)
- \( \angle JGI \cong \angle HIG \)

#### To Prove:
- \( \angle J \cong \angle H \)

#### Proof:

| Statements | Reasons |
|------------|---------|
| 1. \( \overline{JG} \cong \overline{HI} \) | 1. Given |
| 2. \( \angle JGI \cong \angle HIG \) | 2. Given |
| 3. \( \overline{GI} \cong \overline{IG} \) | 3. Reflexive Property of Congruence (Any segment is congruent to itself) |
| 4. \( \triangle JGI \cong \triangle HIG \) | 4. Side-Angle-Side (SAS) Congruence Postulate (Two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle) |
| 5. \( \angle J \cong \angle H \) | 5. Corresponding Parts of Congruent Triangles are Congruent (CPCTC) |

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Problem 2: Prove that \( \triangle PRQ \cong \triangle TRS \)

#### Given:
- \( \angle P \cong \angle T \)
- \( R \) is the midpoint of \( \overline{QS} \)

#### To Prove:
- \( \triangle PRQ \cong \triangle TRS \)

#### Proof:

| Statements | Reasons |
|------------|---------|
| 1. \( \angle P \cong \angle T \) | 1. Given |
| 2. \( R \) is the midpoint of \( \overline{QS} \) | 2. Given |
| 3. \( \overline{QR} \cong \overline{SR} \) | 3. Definition of Midpoint (The midpoint divides a segment into two congruent parts) |
| 4. \( \angle PRQ \cong \angle TRS \) | 4. Vertical Angles Theorem (Vertical angles are always congruent) |
| 5. \( \triangle PRQ \cong \triangle TRS \) | 5. Angle-Side-Angle (ASA) Congruence Postulate (Two angles and the included side of one triangle are congruent to two angles and the included side of another triangle) |

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Problem 3: Prove that \( \overline{AB} \cong \overline{DE} \)

#### Given:
- \( \angle B \cong \angle E \)
- \( \angle BCA \cong \angle ECD \)
- \( C \) is the midpoint of \( \overline{AD} \)

#### To Prove:
- \( \overline{AB} \cong \overline{DE} \)

#### Proof:

| Statements | Reasons |
|------------|---------|
| 1. \( \angle B \cong \angle E \) | 1. Given |
| 2. \( \angle BCA \cong \angle ECD \) | 2. Given |
| 3. \( C \) is the midpoint of \( \overline{AD} \) | 3. Given |
| 4. \( \overline{AC} \cong \overline{DC} \) | 4. Definition of Midpoint (The midpoint divides a segment into two congruent parts) |
| 5. \( \triangle ABC \cong \triangle DEC \) | 5. Angle-Side-Angle (ASA) Congruence Postulate (Two angles and the included side of one triangle are congruent to two angles and the included side of another triangle) |
| 6. \( \overline{AB} \cong \overline{DE} \) | 6. Corresponding Parts of Congruent Triangles are Congruent (CPCTC) |

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Final Answers:

1. \( \boxed{\angle J \cong \angle H} \)
2. \( \boxed{\triangle PRQ \cong \triangle TRS} \)
3. \( \boxed{\overline{AB} \cong \overline{DE}} \)