Question

Prove that a pair of opposite sides in a quadrilateral are equal, then it is a parallelogram ?

Answer 1

Hola Mate!❤

Below is your answer.

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Given,

In a quadrilateral ABCD,

AB = CD , AD = BC.

To prove,

ABCD is a parallelogram.

Construction,

Join AC.

Proof

Consider ΔABC and ΔCDA

AC = AC (common side)

DA = CB (given)

AB = CD (given)

∴ ΔABC ≅ ΔCDA by SSS congruency.

⇒ ∠DCA = ∠CAB (CPCT).

⇒ ∠CAD = ∠ACB (CPCT)

∴ AB║DC (Since alternate angles ∠DCA and ∠CAB are equal)  →  1

∴ DA║CB (Since alternate angles ∠CAD and ∠ACB are equal) → 2

From 1 and 2,

∴ ABCD is a parallelogram as two pairs of opposite sides are parallel.

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Hope It Helps!

Answer 2

Given : ABCD is a quadrilateral

To prove : Oposite sides of quadrilateral ABCD are equal and it is a parallelogram

Construction : Join AC

Proof : In ΔADC & ΔABC,

∠DAC = ∠BCA (pair of alternate interior angles)

∠BAC = ∠ACD (pair of alternate interior angles)

AC = AC (Common)

∴ ΔADC ≅ ΔABC by ASA congruence condition

By CPCT, we can say that

AD = BC and,

AB = DC

∴ ABCD is a parallelogram as its opposite sides are equal

Hope this helps

@PoojaBBSR