Triangle Congruence Proofs Drag and Drop Activity - Free Printable

Let’s solve this step-by-step to complete the proof that △LMO ≅ △NOM.

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Given:

- ∠L ≅ ∠N
- ∠LOM ≅ ∠NMO

We are to prove: △LMO ≅ △NOM

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Step 1: Identify what we already know

We have two pairs of congruent angles:
- ∠L ≅ ∠N (given)
- ∠LOM ≅ ∠NMO (given)

We need a third piece to prove triangle congruence. Since we have two angles, we can try AAS (Angle-Angle-Side) or ASA (Angle-Side-Angle).

Notice that side MO is common to both triangles △LMO and △NOM.

So, MO ≅ MO by the Reflexive Property.

That gives us:

- ∠L ≅ ∠N (given)
- ∠LOM ≅ ∠NMO (given)
- MO ≅ MO (reflexive property)

This is AAS — two angles and a non-included side.

Wait — let’s check: In △LMO and △NOM,

- ∠L and ∠N are corresponding.
- ∠LOM and ∠NMO are corresponding.
- Side MO is opposite ∠L in △LMO and opposite ∠N in △NOM.

Since ∠L ≅ ∠N, and MO is opposite both, then MO is the non-included side for the two angles.

✔ So, AAS is the correct congruence theorem.

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Now, fill in the proof table:

We need to match statements with reasons.

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#### Statement 1:
We start with what’s given.

> ∠L ≅ ∠N

Reason: Given

✔ So:
- Statement 1: `∠L ≅ ∠N`
- Reason 1: `Given`

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#### Statement 2:
Next given:

> ∠LOM ≅ ∠NMO

Reason: Given

✔ So:
- Statement 2: `∠LOM ≅ ∠NMO`
- Reason 2: `Given`

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#### Statement 3:
We need the shared side.

> MO ≅ MO

Reason: Reflexive Property

✔ So:
- Statement 3: `MO ≅ MO`
- Reason 3: `Reflexive Property`

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#### Statement 4:
Now we conclude the triangles are congruent.

> △LMO ≅ △NOM

Reason: AAS

✔ So:
- Statement 4: `△LMO ≅ △NOM`
- Reason 4: `AAS`

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✔ Final Answer:

| STATEMENTS | REASONS |
|------------------------|-------------------|
| 1. ∠L ≅ ∠N | 1. Given |
| 2. ∠LOM ≅ ∠NMO | 2. Given |
| 3. MO ≅ MO | 3. Reflexive Property |
| 4. △LMO ≅ △NOM | 4. AAS |

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📝 Note:

The diagram shows points L, M, N, O forming a quadrilateral with diagonal MO. The triangles being compared are △LMO and △NOM — which share side MO.

Even though the order of vertices in the triangle names might look swapped, as long as the correspondence matches the angles and side, AAS holds.

Also, note that “△NOM” is the same as “△MON” — just written differently. The congruence statement must reflect correct vertex correspondence, but since we’re using AAS with the given angles and shared side, it works.

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✔ All done! The proof is complete.