Practice geometry skills with this worksheet featuring three problems on proving triangle congruence using two-column proofs.

Here's the complete solution with reasons filled in for all three congruent triangle proofs.

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PROOF 1:

Given: $\overline{JG} \cong \overline{HI}$, $\angle JGI \cong \angle HIG$
Prove: $\angle J \cong \angle H$

| 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 |
| 4. $\triangle JGI \cong \triangle HIG$ | 4. SAS (Side-Angle-Side) Congruence Postulate |
| 5. $\angle J \cong \angle H$ | 5. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) |

> Explanation: We are given two sides and the included angle of one triangle congruent to two sides and the included angle of another triangle (SAS). So triangles JGI and HIG are congruent. Therefore, their corresponding angles ∠J and ∠H are congruent.

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PROOF 2:

Given: $\angle P \cong \angle T$, R is the midpoint of $\overline{QS}$
Prove: $\triangle PRQ \cong \triangle TRS$

| 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 |
| 4. $\angle PRQ \cong \angle TRS$ | 4. Vertical Angles Theorem |
| 5. $\triangle PRQ \cong \triangle TRS$ | 5. AAS (Angle-Angle-Side) Congruence Theorem |

> Explanation: We have two angles (∠P ≅ ∠T and vertical angles ∠PRQ ≅ ∠TRS) and a non-included side ($\overline{QR} \cong \overline{SR}$) — this satisfies AAS. So the triangles are congruent.

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PROOF 3:

Given: $\angle B \cong \angle E$, $\angle BCA \cong \angle ECD$, C is the midpoint of $\overline{AD}$
Prove: $\overline{AB} \cong \overline{DE}$

| 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 |
| 5. $\triangle ABC \cong \triangle DEC$ | 5. AAS (Angle-Angle-Side) Congruence Theorem |
| 6. $\overline{AB} \cong \overline{DE}$ | 6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) |

> Explanation: Two angles (∠B ≅ ∠E, ∠BCA ≅ ∠ECD) and a non-included side ($\overline{AC} \cong \overline{DC}$) give us AAS congruence. Once triangles ABC and DEC are congruent, their corresponding sides AB and DE must be congruent.

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✔ All proofs are now complete with correct reasoning steps. Let me know if you’d like diagrams or further explanations!