I"m thinking of a parallelogram whose diagonals room congruent. Name that parallelogram.

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Not every parallelograms have actually congruent diagonals. Rhombuses carry out not have actually congruent diagonals. Rectangles do have congruent diagonals, and so perform squares. You can not conclude that the parallel that I"m reasoning of is a square, though, because that would certainly be too restrictive. As soon as playing ?Name that Quadrilateral,? your answer need to be as basic as possible. Since a square is a rectangle but a rectangle require not it is in a square, the most basic quadrilateral the fits this description is a rectangle.

Theorem 16.5: If the diagonals that a parallelogram space congruent, then the parallelogram is a rectangle.

Figure 16.5 mirrors parallelogram ABCD through congruent diagonals AC and BD. Due to the fact that we are managing a parallelogram, you understand that opposite sides room congruent. You deserve to use the SSS Postulate to display that ?ACD ~= ?DBA. Making use of CPOCTAC, we can show ?A ~= ?D. Since ABCD is a parallelogram, opposite angles space congruent, for this reason ?A ~= ?C and also ?B ~= ?D. Through the transitive home of ~=, you have all 4 angles congruent. Because the measures of the internal angles that a quadrilateral add up to 360, friend can show that all four angles that our parallelogram are ideal angles. That"s more than sufficient to make your parallel a rectangle.

Figure 16.5Parallelogram ABCD v congruent diagonals AC and BD.

StatementsReasons
1. Parallelogram ABCD v AC ~= BCD Given
2. AB ~= CD theorem 15.4
4. ?ACD ~= ?DBA SSS Postulate
5. ?A ~= ?D CPOCTAC
6. ?A ~= ?C and ?B ~= ?D to organize 15.5
7. m?A = m?C and m?B = m?D Definition that ~=
8. m?A + m?B + m?C + m?D = 360 The measures of the interior angles that a quadrilateral include up to 360
9. m?A + m?A + m?A + m?A = 360 Substitution (steps 7 and 8)
10. m?A = 90 Algebra
11. ?A is a right angleDefinition of appropriate angle
12.Parallelogram ABCD is a rectangleDefinition the rectangle

Excerpted native The finish Idiot"s overview to Geometry 2004 through Denise Szecsei, Ph.D.. All rights reserved including the best of reproduction in whole or in part in any form. Provided by setup with Alpha Books, a member that Penguin team (USA) Inc.

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