"The consecutive angles of a parallelogram are supplementary."

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 consecutive angles are supplementary.

Given: Parallelogram ABCD

Prove: Angle BAD and angle CBA are supplementary. *This is a proof for a single pair of consecutive angles. To fully prove the statement above, the 3 other pairs of consecutive angles must also be proved to be supplementary.

• Consider how a parallelogram is constructed------parallel lines.

• Consider properties of parallel lines, linear pairs, and vertical angles.
• Click here to investigate this sketch to help with the steps of the proof.

Proof:

1. Angle BAD is congruent to angle GCB.
2. Angle GBC and angle CBA are supplementary.
3. Angle GBC + angle CBA = 180
4. Angle BAD + angle CBA = 180
5. Angle BAD and angle CBA are supplementary.

Extension:

1. Use the previous proof as an example to prove that the remaining three pairs of consecutive angles of a parallelogram are supplementary.
2. Transform the two-column proof above and in extension 1. into a paragraph proof.

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.