Congruent Triangle Proofs Practice Worksheet

Let's solve each of the congruent triangle proofs step by step, filling in the statements and reasons for each proof. We'll go through them one at a time.

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Problem 1:

Given:
- $ \overline{JG} \cong \overline{HI} $
- $ \angle JGI \cong \angle HIG $

Prove:
- $ \angle J \cong \angle H $

We are given two sides and an included angle, but let’s look carefully.

#### Proof Table:

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

> ✔ Explanation:
We used SAS because:
- $ \overline{JG} \cong \overline{HI} $ (given)
- $ \angle JGI \cong \angle HIG $ (given)
- $ \overline{GI} \cong \overline{IG} $ (reflexive)

So triangles $ \triangle JGI $ and $ \triangle HIG $ are congruent by SAS.

Then, corresponding angles $ \angle J $ and $ \angle H $ are congruent by CPCTC.

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Problem 2:

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

Prove:
- $ \triangle PRQ \cong \triangle TRS $

#### Proof Table:

| 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 are Congruent |
| 5. $ \triangle PRQ \cong \triangle TRS $ | 5. ASA Congruence Postulate (Angle-Side-Angle) |

> ✔ Explanation:
We have:
- $ \angle P \cong \angle T $ (given)
- $ \overline{QR} \cong \overline{SR} $ (from midpoint)
- $ \angle PRQ \cong \angle TRS $ (vertical angles)

So, ASA applies: two angles and the included side are congruent.

Thus, $ \triangle PRQ \cong \triangle TRS $.

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Problem 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} $

#### Proof Table:

| 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. ASA Congruence Postulate |
| 6. $ \overline{AB} \cong \overline{DE} $ | 6. CPCTC |

> ✔ Explanation:
We use ASA:
- $ \angle B \cong \angle E $ (given)
- $ \overline{AC} \cong \overline{DC} $ (from midpoint)
- $ \angle BCA \cong \angle ECD $ (given)

Note: The order matters — we’re matching:
- $ \angle B $ with $ \angle E $
- Side $ \overline{AC} $ with $ \overline{DC} $
- $ \angle BCA $ with $ \angle ECD $

So, ASA applies.

Then, since the triangles are congruent, their corresponding sides $ \overline{AB} $ and $ \overline{DE} $ are congruent by CPCTC.

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✔ Final Summary:

All three proofs rely on triangle congruence postulates (SAS or ASA) and CPCTC.

- Problem 1: SAS → CPCTC
- Problem 2: ASA → CPCTC
- Problem 3: ASA → CPCTC

These are standard geometry proofs involving midpoints, vertical angles, and given congruences.

Let me know if you'd like these written out in paragraph form or need diagrams explained!