Congruent Triangles and CPCTC Proofs Cut and Paste Activity ... - Free Printable

Let's solve each of the congruent triangle proofs step by step, filling in the statements and reasons for each proof. We'll use geometric postulates, theorems, and definitions to justify each step.

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🔷 Problem 1:

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

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

#### ✔ Step-by-step 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 (a segment is congruent to itself) |
| 4. $ \triangle JGI \cong \triangle HIG $ | 4. SAS Congruence Postulate (Side-Angle-Side): Two sides and included angle are congruent |
| 5. $ \angle J \cong \angle H $ | 5. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) |

> ✔ Explanation: We used SAS to prove the triangles congruent, then concluded that corresponding angles $ \angle J $ and $ \angle H $ are congruent.

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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 $

#### ✔ Step-by-step 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 (divides a segment into two congruent parts) |
| 4. $ \angle PRQ \cong \angle TRS $ | 4. Vertical Angles Theorem (vertical angles are congruent) |
| 5. $ \triangle PRQ \cong \triangle TRS $ | 5. ASA Congruence Postulate (Angle-Side-Angle): Two angles and included side are congruent |

> ✔ Explanation: We used ASA: $ \angle P \cong \angle T $, $ \overline{QR} \cong \overline{SR} $, and vertical angles $ \angle PRQ \cong \angle TRS $. This proves 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} $

#### ✔ Step-by-step 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 |
| 5. $ \triangle ABC \cong \triangle DEC $ | 5. ASA Congruence Postulate (Angle-Side-Angle): $ \angle B \cong \angle E $, $ \overline{AC} \cong \overline{DC} $, $ \angle BCA \cong \angle ECD $ |
| 6. $ \overline{AB} \cong \overline{DE} $ | 6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) |

> ✔ Explanation: We used ASA to prove $ \triangle ABC \cong \triangle DEC $, then used CPCTC to conclude that $ AB \cong DE $.

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📌 Summary of Key Concepts Used:

- SAS: Two sides and included angle.
- ASA: Two angles and included side.
- CPCTC: Once triangles are proven congruent, their corresponding parts are congruent.
- Reflexive Property: A segment or angle is congruent to itself.
- Definition of Midpoint: Divides a segment into two equal parts.
- Vertical Angles Theorem: Vertical angles are congruent.

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✔ All three proofs are now complete with correct reasoning. Let me know if you'd like these formatted as a printable worksheet!