"The opposite angles of a parallelogram are congruent."

A Proof Outline
Using Geometer's Sketchpad
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.



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.

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