Using Coordinates in Proofs
Rule

Parallelogram Opposite Sides Theorem

The opposite sides of a parallelogram are congruent.
parallelogram

In respects to the characteristics of the diagram, the following statement holds true.


Two proofs will be provided for this theorem.

Proof

Using Coordinates

This theorem can be proven by placing the parallelogram on a coordinate plane. For simplicity, vertex will be placed at the origin and vertex on the axis.

parallelogram on a coordinate plane
Since lies on the origin, its coordinates are Point is on the axis, meaning its coordinate is Let be the coordinate of Furthermore, let and be the coordinates of
Note that both and lie on the axis. Therefore, is a horizontal segment. Since opposite sides of a parallelogram are parallel, is also a horizontal segment. This means that and have the same coordinate. Let be the coordinate of
parallelogram and its vertices labeled with their coordinates
Next, the coordinate of will be determined. Since and are parallel, they have the same slope. The slope of can be found using the Slope Formula.
The slope of is By following the same procedure, the slope of can be expressed in terms of
Side Endpoints Substitute Simplify
and
and
As it has been previously stated, since and are parallel, their slopes are equal.
The above equation can be solved for
Solve for
The coordinate of is
parallelogram with its vertices in terms of a and b
Finally, by using the Distance Formula, the length of each side of the parallelogram can be calculated. The length of will be calculated first.
By following the same procedure, all the side lengths can be calculated.
Side Endpoints Substitute Simplify
and
and
and
and
By the Transitive Property of Equality, it can be said that and that
By definition of congruent segments, it can be stated that the opposite sides of a parallelogram are congruent.


Proof

Using Congruent Triangles

This theorem can also be proven by using congruent triangles. Consider the parallelogram and its diagonal

parallelogram and one of its diagonals
It can be noted that two triangles are formed with as a common side.
By the definition of a parallelogram, and are parallel. Therefore, by the Alternate Interior Angles Theorem, it can be stated that and that Furthermore, by the Reflexive Property of Congruence, is congruent to itself.
parallelogram and one of its diagonals and the two pair of congruent angles
Consequently, and have two pairs of congruent angles and an included congruent side.
Therefore, by the Angle-Side-Angle Congruence Theorem, and are congruent triangles.
Since corresponding parts of congruent figures are congruent, is congruent to and is congruent to


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