Geometry: When Is a Parallelogram a Rectangle?

When Is a Parallelogram a Rectangle?

I'm thinking of a parallelogram whose diagonals are congruent. Name that parallelogram.

Not all parallelograms have congruent diagonals. Rhombuses do not have congruent diagonals. Rectangles do have congruent diagonals, and so do squares. You cannot conclude that the parallelogram that I'm thinking of is a square, though, because that would be too restrictive. When playing “Name That Quadrilateral,” your answer must be as general as possible. Because a square is a rectangle but a rectangle need not be a square, the most general quadrilateral that fits this description is a rectangle.

  • Theorem 16.5: If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

Figure 16.5 shows parallelogram ABCD with congruent diagonals ¯AC and ¯BD. Because we are dealing with a parallelogram, you know that opposite sides are congruent. You can use the SSS Postulate to show that ΔACD ~= ΔDBA. Using CPOCTAC, we can show ∠A ~= ∠D. Because ABCD is a parallelogram, opposite angles are congruent, so ∠A ~= ∠C and ∠B ~= ∠D. By the transitive property of ~=, you have all four angles congruent. Because the measures of the interior angles of a quadrilateral add up to 360º, you can show that all four angles of our parallelogram are right angles. That's more than enough to make your parallelogram a rectangle.

Figure 16.5Parallelogram ABCD with congruent diagonals ¯AC and ¯BD.

1. Parallelogram ABCD with ¯AC ~= ¯BCD Given
2. ¯AB ~= ¯CD Theorem 15.4
3. ¯AD ~= AD Reflexive property of ~=
4. ΔACD ~= ΔDBA SSS Postulate
5. ∠A ~= ∠D CPOCTAC
6. ∠A ~= ∠C and ∠B ~= ∠D Theorem 15.5
7. m∠A = m∠C and m∠B = m∠D Definition of ~=
8. m∠A + m∠B + m∠C + m∠D = 360º The measures of the interior angles of a quadrilateral add up to 360º
9. m∠A + m∠A + m∠A + m∠A = 360º Substitution (steps 7 and 8)
10. m∠A = 90º Algebra
11. ∠A is a right angleDefinition of right angle
12.Parallelogram ABCD is a rectangleDefinition of rectangle
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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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