Congruent Triangles Notes and Worksheets - Lindsay Bowden - Free Printable

Let's solve each of the congruent triangle proofs step by step, filling in the reasons for each statement. We’ll go through all three problems.

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🔷 Problem 1:

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

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

#### ✔ 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 are given two sides and the included angle between them. The shared side $ \overline{GI} $ is common to both triangles. So, we use SAS to prove the triangles congruent. Then, since corresponding parts of congruent triangles are congruent, $ \angle J \cong \angle H $.

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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, by ASA, the triangles are congruent.

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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 (Angle-Side-Angle) |
| 6. $ \overline{AB} \cong \overline{DE} $ | 6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) |

> ✔ Explanation:
We have:
- $ \angle B \cong \angle E $,
- $ \overline{AC} \cong \overline{DC} $,
- $ \angle BCA \cong \angle ECD $.

This matches ASA: two angles and the included side. So, $ \triangle ABC \cong \triangle DEC $. Therefore, their corresponding sides $ \overline{AB} $ and $ \overline{DE} $ are congruent.

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✔ Final Answers Summary:

#### Problem 1:
- Reasons:
1. Given
2. Given
3. Reflexive Property
4. SAS
5. CPCTC

#### Problem 2:
- Reasons:
1. Given
2. Given
3. Definition of Midpoint
4. Vertical Angles Theorem
5. ASA

#### Problem 3:
- Reasons:
1. Given
2. Given
3. Given
4. Definition of Midpoint
5. ASA
6. CPCTC

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Let me know if you'd like a visual explanation or help with any specific step!