Q1 of 33 Page 108

Let us prove that, if the lengths of two diagonals of a parallelogram are equal, then the parallelogram will be a rectangle.


Consider the parallelogram ABCD with diagonals AC and BD as shown


A parallelogram is a rectangle if its adjacent angles are 90° and opposite sides are equal


Now as it is given that it is a parallelogram which means opposite sides are equal so we need to only check for the adjacent angles


To prove parallelogram ABCD is rectangle we just need to prove the adjacent angles are 90°


Consider ΔBAD and ΔCDA


AC = BD … given


AB = DC … opposite sides of a parallelogram


AD is the common side


Therefore, ΔBAD ≅ ΔCDA … SSS test for congruency


⇒ ∠BAD = ∠CDA …corresponding angles of congruent triangles …(i)


As it is given that ABCD is parallelogram


⇒ ∠BAD + ∠CDA = 180° … sum of adjacent angles of a parallelogram is 180°


Using equation (i)


⇒ ∠CDA + ∠CDA = 180°


⇒ 2 × ∠CDA = 180°


⇒ ∠CDA = 90°


⇒ ∠BAD = 90°


Adjacent angles are 90° implies ABCD is a rectangle


Therefore, if the lengths of two diagonals of a parallelogram are equal, then the parallelogram will be a rectangle.


More from this chapter

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1

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 2

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Let us prove that, if in a parallelogram, the diagonals are equal in lengths and intersect at right angles, the parallelogram will be a square.

 3

Let us prove that, a parallelogram whose diagonals intersect at right angles is a rhombus.