# Conditions for Quadrilateral to be Parallelogram

**Theorem 5**

A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and equal.

**Given:**In Quadrilateral ABCD, AB = DC and AB Ã¯Ã¯ DC.

**To prove:**Quadrilateral ABCD is a parallelogram.

**Construction:**Join B to D.

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**Proof** **:**

In Î”ABD and Î”CDB,

AB = DCÂ Â ...... (given)

BD = BDÂ ...... (common)

âˆ ABD = âˆ BDC ...... (alternate Interior angles of ABÃ¯Ã¯DC)

.^{.}.Â Î”ABD â‰… Î”CDB (SAS Congruence Criteria)

.^{.}.Â âˆ ADB = âˆ CBD

But âˆ ADB and âˆ CBD are alternate interior angles

of AD and BC withÂ transversal BD.

.^{.}.Â Â Â Â Â Â Â AD Ã¯Ã¯ BC

Thus,Â AB Ã¯Ã¯ DC and AD Ã¯Ã¯ BC

.^{.}. Quadrilateral ABCD is a parallelogram.

To summarise: A quadrilateral is a parallelogram if:

(i)Â both pair of opposite sides are equal.

(ii)Â both pair of opposite angles are equal.

(iii) diagonals bisect each other.

(iv) a pair of opposite sides is equal as well asÂ parallel.

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In a parallelogram ABCD, X and Y are the mid-points of sides AB and DCÂ respectively. Show that DXBY is a parallelogram.

**Given: ** In a parallelogram ABCD, X is the mid-point of AB (AX = XB)Â and Y is the mid-point of DC (DY = YC).

**To prove:** DXBY is a parallelogram.

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**Proof:**

AB = DC......(opposite sides of a Ã¯Ã¯^{gm})

Â AB = DC

XB = DYÂ .....(i)

AlsoÂ Â Â Â Â AB Ã¯Ã¯ DC..... (opposite sides of Ã¯Ã¯^{gm})

.^{.}.Â Â Â Â Â Â Â Â XB Ã¯Ã¯ DY...... (ii)

Now in quadrilateral DXBY, we have XB = DYÂ andÂ XB Ã¯Ã¯ DY

.^{.}.DXBY is a parallelogram.

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