Ways to tell if a quadrilateral is a parallelogram (Geometry)

There are several ways to tell if a particular quadrilateral is a parallelogram or not.  The thing that the students MUST NOT do is to try to draw the quadrilateral and state that the quadrilateral is a parallelogram "just because it looks like it".  The students will need to prove using one of the 6 methods to prove that a quadrilateral is a parallelogram.

Given the sides of the quadrilateral, as long as one of the condition is met, it means that the quadrilateral is a parallelogram.  In our example, we will assume we have a parallelogram ABCD:

• Method 1:  If the opposite sides of the quadrilateral are parallel to the other. This means that in a parallelogram there are 2 pairs of parallel sides.  In order to prove that the opposite sides are parallel, you have to prove that the slope of AB is the same as the slope of CD, and the slope of BC is the same as the slope of AD.
  

• Method 2:  If the opposite sides of the quadrilateral are congruent to the other. This means that in a parallelogram there are 2 pairs of congruent sides.  In order to prove that the opposite sides of the parallelogram are congruent, we will have to use the distance formula to show that the lengths of the opposite sides are identical.  In our example, we will need to prove that the length of AB is the same as the length of CD, and the length of BC is the same as the length of AD.
  

• Method 3:  If opposite angles of the quadrilateral are congruent.  This means that there are 2 pairs of congruent angles in a parallelogram.

• Method 4:  If the consecutive angles of the parallelogram are supplementary.

• Method 5:  The diagonals of the parallelogram bisect each other.  In order to prove that the diagonals of the parallelogram bisect each other, we can use the midpoint formula.  Using the midpoint formula, we want to show that the midpoint of AC and the midpoint of BD are the exact same point.  And also by the definition of midpoint formula, it also shows it is the bisector of the diagonals.
  

• Method 6:  Either diagonal of the parallelogram forms 2 congruent triangles.