Proving Lines Parallel with Triangle Congruence SSS SAS AAS and ... - Free Printable

Let’s go step by step to understand why the proof works.

We are told that M is the midpoint of both AD and BC. That means:

- AM = MD (because M cuts AD in half)
- BM = MC (because M cuts BC in half)

Also, look at angles ∠AMB and ∠DMC — they are across from each other where two lines cross (AD and BC intersect at M). These are called vertical angles, and vertical angles are always equal. So ∠AMB ≅ ∠DMC.

Now we have:
- Two sides equal: AM ≅ MD and BM ≅ MC
- The angle between them equal: ∠AMB ≅ ∠DMC

That’s enough to say triangle ABM is congruent to triangle DMC using the SAS (Side-Angle-Side) rule.

Once triangles are congruent, all their matching parts are equal. So angle A (in triangle ABM) matches angle D (in triangle DMC), meaning ∠A ≅ ∠D.

Now here’s the key: if you have two lines (AB and CD) cut by a transversal (AD), and the alternate interior angles (∠A and ∠D) are equal, then the lines must be parallel.

That’s exactly what the Converse of the Alternate Interior Angles Theorem says.

So putting it all together:

1. M is midpoint → gives us equal segments.
2. Vertical angles → gives us equal angles.
3. SAS → proves triangles congruent.
4. CPCTC → gives us equal corresponding angles (∠A ≅ ∠D).
5. Converse of Alt. Int. Angles Thm → proves lines are parallel.

Final Answer:
The proof is complete and correct as shown in the table. Each statement follows logically from the previous ones using valid geometric reasons. Therefore, AB is parallel to CD.