Opposite Sides and Angles are Congruent

Our goal is to prove that

• • opposite sides of a parallelogram are congruent, and

  • opposite angles of a parallelogram are congruent

Start with a basic parallelogram

Now we're going to extend these lines a bit. Also, you may have noticed that we love to make our classic "two parallel lines and a transversal", so we're going to add a transversal too.

That's a lot of lines, so we're just going to worry about the ones we care about first. I'll go ahead and grey out the ones we're not using right now.

For now, we're going to focus on lines AB and CD, and the transversal

Next, we'll go ahead and temporarily erase everything else

At last, our classic "two parallel lines and a transversal". We can see here that 2 and 5 are alternate interior angles.

All alternate interior angles are congruent, so ∠2 ≅ ∠5

Coloring in those angles, here is our updated parallelogram

Now we'll focus on the other pair of parallel lines

Now, we're going to focus on lines AC and BD, and the transversal

We'll go ahead and temporarily erase everything else

Yippie! Our classic "two parallel lines and a transversal"! We can see here that 3 and 4 are alternate interior angles.

All alternate interior angles are congruent, so ∠3 ≅ ∠4

Coloring in those angles, here is our updated parallelogram

We can use this picture for the rest of the proof. Note that if you are good at seeing transversals, you could actually just start with this picture.

We now have two pairs of congruent angles, with a shared side between them. This lets us use ASA for triangle congruence.

• ΔABC ≅ ΔDCB

Now its just a simple matter of Corresponding Parts of Congruent Triangles are Congruent.

• AB ≅ DC

• CA ≅ BD

And with that we've proven that opposite sides of a parallelogram are congruent

Now for the opposite angles. For top left and bottom right, we can use CPCTC,

• ∠1 ≅ ∠6

For bottom left and top right, we can just use the Angle Sum Postulate

• m∠Bottom Left = m∠3 + m∠5

• m∠Top Right = m∠4 + m∠2

But since ∠3 ≅ ∠4 and ∠2 ≅ ∠5, then m∠3 + m∠5 = m∠4 + m∠2

• ∠Bottom Left ≅ ∠Top Right

And that's everything.

Below is a 2-column version of this proof

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2-column Proof

Prove that in a parallelogram,

1. opposite sides are congruent

2. opposite angles are congruent

To watch a Khan Academy video of these proofs, click here for sides or here for angles