Triangle Congruence Proofs worksheet - Free Printable

Let’s solve both proofs step by step.

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First Proof: Proving △EFG ≅ △GHE

We are given:
- EF ≅ GH
- FG ≅ HE

And we need to prove: △EFG ≅ △GHE

Looking at the diagram, both triangles share side GE. So:

Statement 1: EF ≅ GH → Reason: Given (this was told to us)

Statement 2: FG ≅ HE → Reason: Given (also told to us)

Statement 3: GE ≅ GE → Reason: Reflexive Property (any segment is congruent to itself — this is the shared side)

Now we have all three sides of one triangle congruent to all three sides of the other triangle:
- EF ≅ GH
- FG ≅ HE
- GE ≅ GE

That means we can use SSS (Side-Side-Side) to say the triangles are congruent.

So Statement 4: △EFG ≅ △GHE → Reason: SSS

✔ First proof done!

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Second Proof: Proving △BCA ≅ △DCE

Given:
- BC ≅ DC
- AC ≅ EC

Prove: △BCA ≅ △DCE

Look at the diagram — point C is where the two lines cross. That means angles ∠ACB and ∠ECD are vertical angles. Vertical angles are always congruent.

So let’s fill in the table:

Statement 1: We’re told BC ≅ DC and AC ≅ EC → so write that as:
BC ≅ DC ; AC ≅ EC → Reason: Given

Statement 2: The angle between those sides — ∠ACB and ∠ECD — are vertical angles → so:
∠ACB ≅ ∠ECD → Reason: Vertical Angles Theorem

Now we have:
- Two sides congruent: BC ≅ DC and AC ≅ EC
- And the included angle between them congruent: ∠ACB ≅ ∠ECD

That’s SAS (Side-Angle-Side)!

So Statement 3: △BCA ≅ △DCE → Reason: SAS

✔ Second proof done!

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Final Answer:

First Proof Reasons:
1. Given
2. Given
3. Reflexive Property
4. SSS

Second Proof Statements & Reasons:
1. BC ≅ DC ; AC ≅ EC
2. ∠ACB ≅ ∠ECD
3. SAS