Geometry worksheet titled 'Congruent Triangle Proofs Practice' featuring three problems where students complete two-column proofs.

Here is the completed solution for all three congruent triangle proofs, with explanations for each step.

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## Proof 1: Prove ∠J ≅ ∠H

Given:
- JG ≅ HI
- ∠JGI ≅ ∠HIG

Prove: ∠J ≅ ∠H

| Statements | Reasons |
|-------------------------------|------------------------------------------|
| 1. JG ≅ HI | 1. Given |
| 2. ∠JGI ≅ ∠HIG | 2. Given |
| 3. GI ≅ IG | 3. Reflexive Property of Congruence |
| 4. ΔJGI ≅ ΔHIG | 4. SAS (Side-Angle-Side) Congruence |
| 5. ∠J ≅ ∠H | 5. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) |

✔ Explanation:
- We are given two sides and the included angle? Actually, we have side JG, angle ∠JGI, and side GI in triangle JGI.
- In triangle HIG, we have side HI (given ≅ JG), angle ∠HIG (given ≅ ∠JGI), and side IG (same as GI — reflexive).
- So we have SAS: Side (JG ≅ HI), Angle (∠JGI ≅ ∠HIG), Side (GI ≅ IG).
- Therefore, triangles JGI and HIG are congruent by SAS.
- Then, corresponding angles ∠J and ∠H must be congruent by CPCTC.

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## Proof 2: Prove ΔPRQ ≅ ΔTRS

Given:
- ∠P ≅ ∠T
- R is the midpoint of QS

Prove: ΔPRQ ≅ ΔTRS

| Statements | Reasons |
|-------------------------------|------------------------------------------|
| 1. ∠P ≅ ∠T | 1. Given |
| 2. R is the midpoint of QS | 2. Given |
| 3. QR ≅ SR | 3. Definition of Midpoint |
| 4. ∠PRQ ≅ ∠TRS | 4. Vertical Angles Theorem |
| 5. ΔPRQ ≅ ΔTRS | 5. AAS (Angle-Angle-Side) Congruence |

✔ Explanation:
- We are given ∠P ≅ ∠T.
- Since R is the midpoint of QS, QR ≅ SR (by definition of midpoint).
- Angles ∠PRQ and ∠TRS are vertical angles (formed by intersecting lines PT and QS), so they are congruent.
- Now we have:
- ∠P ≅ ∠T (given),
- ∠PRQ ≅ ∠TRS (vertical angles),
- QR ≅ SR (side opposite one of the angles — note: this is the non-included side).
- This fits AAS (Angle-Angle-Side) congruence criterion.
- Thus, triangles PRQ and TRS are congruent by AAS.

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## Proof 3: Prove AB ≅ DE

Given:
- ∠B ≅ ∠E
- ∠BCA ≅ ∠ECD
- C is the midpoint of AD

Prove: AB ≅ DE

| Statements | Reasons |
|-------------------------------|------------------------------------------|
| 1. ∠B ≅ ∠E | 1. Given |
| 2. ∠BCA ≅ ∠ECD | 2. Given |
| 3. C is the midpoint of AD | 3. Given |
| 4. AC ≅ DC | 4. Definition of Midpoint |
| 5. ΔABC ≅ ΔDEC | 5. AAS (Angle-Angle-Side) Congruence |
| 6. AB ≅ DE | 6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) |

✔ Explanation:
- We are given two pairs of angles: ∠B ≅ ∠E and ∠BCA ≅ ∠ECD.
- Since C is the midpoint of AD, AC ≅ DC (definition of midpoint).
- In triangle ABC and triangle DEC:
- ∠B ≅ ∠E (given),
- ∠BCA ≅ ∠ECD (given),
- AC ≅ DC (side opposite ∠B and ∠E respectively — non-included side).
- This satisfies AAS congruence.
- Therefore, ΔABC ≅ ΔDEC.
- By CPCTC, corresponding sides AB and DE are congruent.

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## ✔ Final Answer Summary:

Proof 1 Reasons:

1. Given
2. Given
3. Reflexive Property
4. SAS
5. CPCTC

Proof 2 Reasons:

1. Given
2. Given
3. Definition of Midpoint
4. Vertical Angles Theorem
5. AAS

Proof 3 Reasons:

1. Given
2. Given
3. Given
4. Definition of Midpoint
5. AAS
6. CPCTC

These proofs use standard triangle congruence criteria (SAS, AAS) and properties like reflexive, midpoint, and vertical angles to logically deduce the required conclusions.