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.**

- Angle DCA is congruent to angle BAC.
- Angle ADC is congruent to angle CBA.
- Triangle ADC is congruent to triangle CBA.
- Segment AD is congruent to segment BC.

**Then we will look at both diagonals.**

- Angle BAD is congruent to angle CBA.
- Segment AB is congruent to segment AB.
- Triangle DAB is congruent to triangle CBA.
**Segment AC is congruent to segment BD.**

**Extension****:**

- Transform the two-column proof into a paragraph proof.
- 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**.

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