Question

prove that a quadrilateral is a parallelogram, if its one pair of opposite sides are equal and parallel.​

Answer 1

$\:\:\:\:\huge\bullet\:\:\:\:\:\sf\red{Perfect\:Answer}$

Answer 2

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$\huge{\underline{\underline{\sf{\orange{</p><p>Question:-}}}}}$

prove that a quadrilateral is a parallelogram, if its one pair of opposite sides are equal and parallel.

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${\blue{\sf\underline{Given}}}$

A quad. ABCD in which AB = DC Nd AB || DC.

${\green{\sf\underline{To\:prove}}}$

ABCD is a ||gm.

${\blue{\sf\underline{Construction}}}$

Join A and C.

${\green{\sf\underline{Proof}}}$

In ∆ABC and CDA,we have

$AB = DC \: \: (given) \: \: \: \: \: \: \: AC = CA \: \: (commom)$

And ∠BAC = ∠DCA [alt.interior,as AB || DC and CA cuts them.]

∴ ∆ABC ≅ ∆CDA (SAS-criteria)

∴∠BCA = ∠DAC. (c.p.c.t.).

But,these Re alternate interior angles.

∴AD || BC.

Now,AB || DC and AD || BC.

∴ABCD is a ||gm.

${\blue{\sf\underline{Summary}}}$

A quadrilateral is a parallelogram

(i) if both pairs of opposite sides are equal

or (ii) if both pairs of opposite angles are equal

or (iii) if the diagonals bisect each other

or (iv) if a pair of opposite sides are equal and parallel.

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