"The opposite sides of a parallelogram are congruent."

A Proof Outline
by David Wise

Note: I recommend that this page be printed out, so that the instructions are easier to follow.

In order to successfully complete a proof, it is important to think of the definition and the construction of a parallelogram.
In the following outline, I will provide the statements, you provide the reasons.

Prove: If a quadrilateral is a parallelogram, then the opposite sides are congruent.

Given: Parallelogram ABCD

Prove: Segment AD is congruent to segment BC and segment AB is congruent to segment DC.

• Consider how a parallelogram is constructed------parallel lines.
• Consider properties of parallel lines.
• Construct a diagonal of the parallelogram to create 2 triangles.
• Consider triangle congruency properties.

• Click here to investigate this sketch to help with the steps of the proof.

Proof:

• Angle DBA is congruent to angle BDC.

• Angle ADB is congruent to angle CBD.

• Triangle ABD is congruent to triangle CDB.
• Segment AD is congruent to segment BC and segment AB is congruent to segment DC.

Extension:

1. Transform the two-column proof into a paragraph proof.
2. Find an alternative way to prove that the opposite side of a parallelogram are congruent.

If you have any questions while trying to complete this investigation, or suggestions that would be useful, especially for use at the high school level, please send e-mail to esiwdivad@yahoo.com.