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If the diagonals of parallelogram are …

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If the diagonals of parallelogram are equal then show that it is a rectangula

Posted by Mohit Kumar 8 years, 4 months ago

CBSE > Class 09 > Mathematics

2 answers

Mona Chaudhary 8 years, 4 months ago

Let take parallelogram ABCD AC=BD given Take triangle ACB and BDA. AB=AB common. AC = BD given AD = BC (oposite side are equal) ACB congurent BDA by SSS. Angle A = B by cpct A+B= 180 ( the sum of suplemantey angle =180) A+A =180. 2A=180. A=180/2. A=90. A+B=180. 90+B=180. B=90. C=B (oposite angle are equal) C=90. A=D ( oposite angle are equal) D=90. ABCD is a rectangle hence prove Because all angle of 90

Anima Singh 8 years, 4 months ago

Draw a parallelogram ABCD with equal diagonals AC and BD. Now in ∆ABC and ∆DBC , show them congruent. By AB =DC ( opposite sides of parallelogram) , AC = BD ( as given equal diagonals) and BC as common side. Then by SSS congruence rule ∆ABC and ∆DBC become congruent. Then angle ABC and angle DCB are equal by cpct. And as per the property of parallelogram, sum of adjacent angles is180°, so angle ABC + angle DCB =180° Now 2angle ABC = 180 ( ABC and DCB are equal) Angle ABC= 90° And as per the property of rectangle if one angle of a quadrilateral is 90 then it is a rectangle. Hence proved !

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