Question

Show that of the diagonals of a parallelogram are equal then show that it is a rectangle

Answer 1

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Answer 2

Question :-

1.Show that of the diagonals of a parallelogram are equal then show that it is a rectangle.

Solution :-

Let ABCD is a parallelogram (llgm)

such that,

AC = BD

In ∆ABC and ∆DCB,

• AC = DB [Given]
• AB = DC [Opposite sides of a parallelogram]
• BC = CB [Common]

∴ ∆ABC ≅ ∆DCB [By SSS congruency]

⇒ ∠ABC = ∠DCB [By C.P.C.T.] …(1)

Now, AB || DC and BC is a transversal.

[ ∵ ABCD is a parallelogram]

∴ ∠ABC + ∠DCB = 180° … (2)

[Co-interior angles]

From (1) and (2), we have

∠ABC = ∠DCB = 90°

i.e., ABCD is a parallelogram having an angle equal to 90°.

∴ ABCD is a rectangle.