Q9. If the diagonals of a parallelogram are equal in length, show that the parallelogram is a rectangle.
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Given: ABCD is a parallelogram whose diagonals are AC and BD and AC = BD.
Need to prove: ABCD is a rectangle
⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━⠀⠀⠀⠀⠀
☯︎ In ∆ABC and ∆ABD,
AB = AB (Common)
BC = AD (Opposite sides of a parallelogram)
AC = BD (Given)
∴ ∆ABC ≅ ∆ABD (SSS axiom)
So, ∠ABC = ∠BAD (c.p.c.t)
But, ∠ABC + ∠BAD = 180° (Co-interior angles)
∴ ∠ABC = ∠BAD = 90°
Hence, a parallelogram whose each angles are 90° is a rectangle.
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