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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.

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.

AB = DC, BC = AD                               [by supposition]

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

∴ ∠B = ∠D     ——————(ii)

AB = DC, BC = AD                               [by supposition]

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

∴ ∠A = ∠C ——————————–(iv)

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.

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

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)

## If two triangles have the three sides of the one equal to the three sides of the other, each to each, then they are equal in all respects.

If two triangles have the three sides of the one equal to the three sides ...