Geometry Worksheet Congruent Triangles - Free Printable

Here is the two-column proof for Problem 1, as requested. Since you uploaded an image with multiple problems, I’ll solve the first one as a model. If you’d like proofs for others, let me know!

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Problem 1:

Given:
- $\overline{AB} \parallel \overline{CD}$
- $\overline{AB} \cong \overline{CD}$

Prove: $\triangle ABC \cong \triangle DCB$

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## ✔ Two-Column Proof:

| Statement | Reason |
|---------------------------------------------------|------------------------------------------------|
| 1. $\overline{AB} \parallel \overline{CD}$ | Given |
| 2. $\angle ABC \cong \angle DCB$ | Alternate Interior Angles Theorem (since $AB \parallel CD$, transversal $BC$) |
| 3. $\overline{AB} \cong \overline{CD}$ | Given |
| 4. $\overline{BC} \cong \overline{CB}$ | Reflexive Property of Congruence |
| 5. $\triangle ABC \cong \triangle DCB$ | SAS (Side-Angle-Side) Congruence Postulate |

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## 🔍 Explanation:

We are given that segment AB is parallel to segment CD and that they are congruent. We are to prove that triangles ABC and DCB are congruent.

- Step 1 & 2: Because AB is parallel to CD, and BC is a transversal crossing them, the alternate interior angles (∠ABC and ∠DCB) are congruent.
- Step 3: We’re directly told AB ≅ CD.
- Step 4: Side BC is common to both triangles — so it’s congruent to itself (reflexive property).
- Step 5: Now we have two sides and the included angle equal in both triangles:
- AB ≅ CD (side)
- ∠ABC ≅ ∠DCB (included angle)
- BC ≅ CB (side)

→ This satisfies the SAS (Side-Angle-Side) congruence criterion.

✔ Therefore, $\triangle ABC \cong \triangle DCB$ by SAS.

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Let me know if you want proofs for Problems 2 through 10 as well — I can provide all of them in the same format!