"The opposite angles 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 angles are congruent.

Given: Parallelogram ABCD Prove: Angle BAD is congruent to angle DCB and angle CBA is congruent to angle ADC.

• 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 CDJ.
2. Angle CDJ is congruent to angle DCB.
3. Angle BAD is congruent to angle DCB.
4. Angle CBA is congruent to angle ICD.
5. Angle ICD is congruent to angle ACD.
6. Angle CBA is congruent to angle ACD.

Extension:

1. Transform the two-column proof into a paragraph proof.
2. Find an alternative way to prove that the opposite angles 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.