Properties of Parallelograms: Sides and Angles

 

A parallelogram is a type of quadrilateral whose pairs of opposite sides are parallel.

Quadrilateral ABCD is a parallelogram because AB = DC and AD = BC.

In order to determine whether a quadrilateral is a parallelogram, we will use properties of quadrilaterals.

 

If a quadrilateral is a parallelogram, then.

  • Opposite sides are congruent,
  • Opposite angles are congruent, and
  • Consecutive angles are supplementary.

If one angle of the parallelogram is a right angle, then all the other angles are right angles too. Consider the figure below.

Its given that J is a right angle, we can also determine that L is a right angle since the opposite sides of parallelograms are congruent. Together, the sum of the measure of those angles is 180 because

We also know that the remaining angles must be congruent because they are also opposite angles.

As per the  Polygon Interior Angles Sum Theorem, we know that all quadrilaterals have angle measures that add up to 360. Since J and L sum up to 180, we know that the sum of K and M will also be 180:

Since K and M are congruent, we can define their measures with the same variable, x. So we have

Therefore, we know that K and M are both right angles. So, we arrive at

Example:

Given that QRST is a parallelogram, find the values of x and y in the diagram below.

Solution:

Since opposite sides of parallelograms are congruent, we have set the quantities equal to each other and solve for x:

Now that we've determined that the value of x is 7, we can use this to plug into the expression given in R. We know that R and T are congruent, so we have

Substitute x for 7 and we get

So, we've determined that x=7 and y=8.