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Home / Geometry / Prove that, if the opposite sides of a quadrilateral are equal and parallel, it is a parallelogram.

Prove that, if the opposite sides of a quadrilateral are equal and parallel, it is a parallelogram.

General enunciation: We have to prove that if the opposite sides of a quadrilateral are equal and parallel, it is a parallelogram.

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Particular enunciation: Let the opposite sides of the quadrilateral ABCD are equal and parallel i.e., BC = AD, BC || AD and AB = CD, AB || CD.

We have to prove that ABCD is a parallelogram.

Construction: We join A, C and B, D.

Proof: In ABC and ADC.

AB = DC, BC = AD                               [by supposition]

and AC = AC ——————(i)         [common side]

ABC ≅ ADC.

∴ ∠B = ∠D     ——————(ii)

In BAD and BCD

AB = DC, BC = AD                               [by supposition]

and BD = BD            ——————–(iii)                   [common side]

BAD ≅ BCD.

∴ ∠A = ∠C ——————————–(iv)

In the quadrilateral ABCD,

BC = AD, AB = DC [ from (i) and (iii)]

and ∠B = ∠D, ∠A = ∠C   [from (ii) and (iv)]

We know that, the opposite sides and angles of a parallelogram are equal.

∴ ABCD is a parallelogram.                      (Proved).

Problem: The median BO of ABC is produced up to D so that BO = OD. Prove that ABCD is a parallelogram.

q1

Solution: General Enunciation: Problem: The median BO of ABC is produced up to D so that BO = OD. Prove that ABCD is parallelogram.

q1

Particular enunciation: Given that the median BO of ABC is produced up to D so that BO = OD. We join C, D and A, D. We have to prove that ABCD is parallelogram.

Proof: In ABC, the point of the side AC is O and given that BO is the median of ABC.

∴ CO = OA   ————————(i)

The diagonal of the quadrilateral ABCD is AC and BD where

CO = OA,                    [ from (i)]

BO = OD.                   [According to construction]

i.e., the diagonals bisect each other.

Therefore, ABCD is a parallelogram.                                     (Proved)

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