This is the last round the ?Name the Quadrilateral.? I"m reasoning of a parallelogram with congruent perpendicular diagonals. Name that parallelogram.
You are watching: If a quadrilateral is a square, then it is a parallelogram
Well, if a parallelogram has actually congruent diagonals, you know that it is a rectangle. If a parallelogram has actually perpendicular diagonals, you understand it is a rhombus. For this reason I"m reasoning of a parallel that is both a rectangle and a rhombus. The only parallelogram the satisfies that description is a square.Theorem 16.8: If the diagonals that a parallelogram room congruent and also perpendicular, the parallelogram is a square.
There"s not lot to this proof, because you"ve done many of the work in the last 2 sections. The drawing in number 16.8 shows a parallelogram through congruent perpendicular diagonals, however it is misleading in that it walk not rather look favor a square. You won"t it is in fooled through the picture, yet you will extract the important information. Special, you require a parallelogram with diagonals AC and also BD that are both perpendicular and congruent. The factor I intentionally attracted a share parallelogram rather than a square is that I desire to be mindful not to i think what ns am trying to prove. If I drew a square, I might be tempted to attract conclusions about the lengths of the nearby sides. As mathematicians in training, it is crucial that you continue to be as much away from the pitfall of assuming what you space trying to prove. It"s not unusual for human being to reap the scenery on your geometric trip and forget come watch where they walk!
Figure 16.8Parallelogram ABCD v AC ~= BD and AC ? BD
|1.||Parallelogram ABCD v AC ~= BD and also AC ? BD||Given|
|2.||Parallelogram ABCD is a rectangle||Theorem 16.5|
|3.||Parallelogram ABCD is a rhombus||Theorem 16.6|
|4.||Parallelogram ABCD is a rectangle v congruent surrounding sides||Definition of a rhombus|
|5.||Parallelogram ABCD is a square||Definition the a square|
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Excerpted from The complete Idiot"s guide to Geometry 2004 by Denise Szecsei, Ph.D.. All civil liberties reserved including the appropriate of reproduction in totality or in component in any type of form. Provided by arrangement with Alpha Books, a member the Penguin team (USA) Inc.
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