Geometry: When Is a Parallelogram a Square?

When Is a Parallelogram a Square?

This is the last round of “Name That Quadrilateral.” I'm thinking of a parallelogram with congruent perpendicular diagonals. Name that parallelogram.

Well, if a parallelogram has congruent diagonals, you know that it is a rectangle. If a parallelogram has perpendicular diagonals, you know it is a rhombus. So I'm thinking of a parallelogram that is both a rectangle and a rhombus. The only parallelogram that satisfies that description is a square.

  • Theorem 16.8: If the diagonals of a parallelogram are congruent and perpendicular, the parallelogram is a square.

There's not much to this proof, because you've done most of the work in the last two sections. The drawing in Figure 16.8 shows a parallelogram with congruent perpendicular diagonals, but it is misleading in that it does not quite look like a square. You won't be fooled by the picture, but you will extract the important information. Specifically, you need a parallelogram with diagonals ¯AC and ¯BD that are both perpendicular and congruent. The reason I intentionally drew a generic parallelogram rather than a square is that I want to be careful not to assume what I am trying to prove. If I drew a square, I might be tempted to draw conclusions about the lengths of the adjacent sides. As mathematicians in training, it is important that you stay as far away from the pitfall of assuming what you are trying to prove. It's not uncommon for people to enjoy the scenery on their geometric trip and forget to watch where they walk!

Figure 16.8Parallelogram ABCD with ¯AC ~= ¯BD and ¯AC ⊥ ¯BD

 Statements Reasons
1. Parallelogram ABCD with ¯AC ~= ¯BD and ¯AC ⊥ ¯BD Given
2. Parallelogram ABCD is a rectangleTheorem 16.5
3. Parallelogram ABCD is a rhombusTheorem 16.6
4. Parallelogram ABCD is a rectangle with congruent adjacent sidesDefinition of a rhombus
5. Parallelogram ABCD is a squareDefinition of a square

That wraps up this episode of “Name That Quadrilateral.” Contestants earned a solid education, which can be passed on to friends, family, and anyone willing to listen. If you plan to be in the area and would like tickets to the taping of “Name That Quadrilateral,” send a postcard to the address at the end of the section with your name, grade level, mathematical interests, and dates that you'll be in the area. See you next time!

Excerpted from The Complete Idiot's Guide to Geometry © 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.

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