Proving a quadrilateral is a parallelogram

key notes :

A parallelogram is a quadrilateral with both pairs of opposite sides parallel. There are several methods to prove that a given quadrilateral is a parallelogram:

Definition – Based Proof

  • A quadrilateral is a parallelogram if both pairs of opposite sides are parallel.
  • Use properties of parallel lines and transversals to verify.

Properties to Prove a Quadrilateral is a Parallelogram

A quadrilateral is a parallelogram if any one of the following conditions is met:

✅ Both pairs of opposite sides are congruent

  • If AB ≅ CD and BC ≅ AD, then ABCD is a parallelogram.

✅ Both pairs of opposite angles are congruent

  • If ∠A ≅ ∠C and ∠B ≅ ∠D, then ABCD is a parallelogram.

✅ Diagonals bisect each other

  • If the diagonals AC and BD bisect each other at M, then ABCD is a parallelogram (AM = MC and BM = MD).

✅ One pair of opposite sides is both parallel and congruent

  • If AB ≅ CD and AB ∥ CD, then ABCD is a parallelogram.

✅ Consecutive angles are supplementary

  • If ∠A + ∠B = 180° and ∠C + ∠D = 180°, then ABCD is a parallelogram.

Methods of Proof

Coordinate Geometry:

  • Use the distance formula to check if opposite sides are congruent.
  • Use the slope formula to check if opposite sides are parallel.
  • Use the midpoint formula to check if diagonals bisect each other.

Two-Column or Paragraph Proofs:

  • Use given information and geometric theorems to logically prove that a quadrilateral is a parallelogram.

Learn with an example

Can you show that this quadrilateral is a parallelogram?

  • yes
  • no

One pair of opposite angles measures 118°. Another angle measures 62°.

The quadrilateral is a parallelogram if the unlabeled angle also measures 62°. Call the unlabeled angle measure x. Set the sum of the interior angle measures equal to 360° and solve for x.

118°+62°+118°+x = 360°

298°+x = 360° Ad

x = 62° Subtract 298° from both sides

Since x=62°, both pairs of opposite angles are congruent. So, this quadrilateral is a parallelogram.

Can you show that this quadrilateral is a parallelogram?

  • yes
  • no

One pair of opposite angles measures 69°. Another angle measures 111°.

The quadrilateral is a parallelogram if the unlabeled angle also measures 111°. Call the unlabeled angle measure x. Set the sum of the interior angle measures equal to 360° and solve for x.

69°+111°+69°+x = 360°

249°+x = 360° Add

x = 111° Subtract 249° from both sides

Since x=111°, both pairs of opposite angles are congruent. So, this quadrilateral is a parallelogram.

Can you show that this quadrilateral is a parallelogram?

  • yes
  • no

Since one diagonal is split into non-congruent segments with length 39 and length 38, the diagonals do not bisect each other.

So, the quadrilateral is not a parallelogram.

🏋️‍♂️ Work it out🏋️‍♀️

Not feeling ready yet? These can help:

🥏Properties of parallelograms

Let’s practice!