Parallelogram

Guide to Understanding Parallelogram Properties, Proofs, and Example Calculations

parallelogramWhat is a Parallelogram?

Contents

Definition: A parallelogram is a quadrilateral, four sided figure, that has opposite sides that are both congruent and parallel.

What does a parallelogram look like? Specific types of parallelograms include rectangles and squares. These shapes are often seen and used in the world around us.

Let’s look at what makes a parallelogram and how to prove them mathematically.


Properties of Parallelograms and Theorems

When GIVEN a parallelogram Theorems for sides, angles, and diagonals

How to prove a quadrilateral is a parallelogram.

If you are given a proof that includes a diagram and it is given to be a parallelogram, one or more of the properties may be used:

  • If a quadrilateral is a parallelogram, the opposite sides are parallel. When opposite sides are parallel then we know that opposite sides have the same slope.
  • If a quadrilateral is a parallelogram, the opposite sides are congruent. This means that the opposite sides are equal in length. This means that the distance between the vertices is equal.
  • If a quadrilateral is a parallelogram, the opposite angles are congruent. This can be used to find the measure of a missing angle.
  • If a quadrilateral is a parallelogram, the consecutive angles are supplementary. This means that the angles add up to be 180 degrees. This can be used to find missing angles
  • If a quadrilateral is a parallelogram, the diagonals bisect each other. This property can be used in unison with the midpoint theorem. It is also used to find lengths of the diagonals. A more complicated proof may use this property to find or prove triangles that are formed in the parallelogram by the diagonals. Using other theorems can prove triangles congruent.

When trying to PROVE a parallelogram Theorems for sides, angles, and diagonals

Are the diagonals of a parallelogram congruent?

If you are trying to prove a figure is a parallelogram any one of the following theorems can be used.

  • If both pairs of opposite sides of a quadrilateral are parallel, the quadrilateral is a parallelogram.
  • If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.
  • If both pairs of opposite angles of a quadrilateral are congruent, the quadrilateral is a parallelogram.
  • If consecutive angles of a quadrilateral are supplementary, the quadrilateral is a parallelogram.
  • If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram.

When trying to prove a shape is a parallelogram, start by looking at what information is given to you. Some information might be in the given section, like parallel lines or congruent measures. Information may be hidden in other shapes. If triangles are inscribed in a quadrilateral, you may have to look at those properties, such as isosceles triangles have a pair of congruent sides. Often the properties from parallel lines that are cut with a transversal are needed to find parallel sides.

With all types of proofs the general idea is to look at what information is given to you, what do you need to prove, and what information is missing to get you there.


Examples of Parallelogram Proofs

How to prove a parallelogram

parallelogram-definition

With this example of a proof, we know that the diagram is a parallelogram. This gives us all of the theorems mentioned earlier. Since we need to prove that two triangles are congruent, we need to look at what we need in order to do that, congruent sides and angles. Here are the theorems and how they assist us in finding the proof.

 

If a quadrilateral is a parallelogram, the opposite angles are congruent.  

parallelogram-properties

If a quadrilateral is a parallelogram, the opposite sides are congruent. parallelogram-congruent-diagonals-example

 

We are now able to prove the triangles are congruent by SAS. Note that there are other ways to go about proving the angles are congruent, this is one example.

parallelogram-theorem

With this example of a proof, we need to show that the given quadrilateral is a parallelogram. Most proofs do not require proving that it is a quadrilateral since it has four sides.

Since one of the givens is showing us that a pair of opposite sides are parallel, we want to start by trying to prove the other pair of sides parallel. Here are the theorems and how they assist us in finding the proof.

parallelogram-proofs If two lines are cut by a transversal and the alternate interior angles are congruent, the lines are parallel.
*This should be known from previous knowledge
parallelogram-quadrilateral-proof If both pairs of opposite sides of a quadrilateral are parallel, the quadrilateral is a parallelogram.