"The diagonals of a rectangle 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 rectangle.
In the following outline, I will provide the statements, you provide the reasons.

Prove: If a quadrilateral is a rectangle, then the diagonals of the rectangle are congruent.

Given: Rectangle ABCD with diagonals AC and BD. Prove: Segment AC is congruent to segment BD.

• A rectangle is a parallelogram, so the definition and properties of a parallelogram apply to a rectangle.
• Consider how a rectangle is constructed------parallel lines.
• Consider properties of parallel lines.
• Consider triangle congruency properties.
• Click here to investigate this sketch to help with the steps of the proof.

Proof:

First we will look at only one diagonal. 1. Angle DCA is congruent to angle BAC.
2. Angle ADC is congruent to angle CBA.
3. Triangle ADC is congruent to triangle CBA.
4. Segment AD is congruent to segment BC.

Then we will look at both diagonals. 1. Angle BAD is congruent to angle CBA.
2. Segment AB is congruent to segment AB.
3. Triangle DAB is congruent to triangle CBA.
4. Segment AC is congruent to segment BD.

Extension:

1. Transform the two-column proof into a paragraph proof.
2. Find an alternative way to prove that the diagonals of a rectangle 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.