In and , AB = DE, , BC = EF and . Vertices A, B and C are joined to vertices D, E and F respectively (See figure). Show that

**(i)** quadrilateral ABED is a parallelogram

**(ii)** quadrilateral BEFC is a parallelogram

**(iii)** and AD = CF

**(iv)** quadrilateral ACFD is a parallelogram.

**(v)** AC=DF

**(vi)**

**Solution (i)**

We have AB=DE and **(Given)**

Therefore, quadrilateral ABED is a parallelogram.

**(A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.)**

**Solution (ii)**

We have BC=EF and **(Given)**

Therefore, quadrilateral BEFC is a parallelogram.

**(A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.)**

**Solution (iii)**

In **solution (i)**, we have showed that ABED is a parallelogram.

and ** (Opposite sides of parallelogram are equal and parallel) (1)**

In **solution (ii)**, we have showed that BEFC is a parallelogram.

and ** (Opposite sides of parallelogram are equal and parallel) (2)**

From **(1)** and **(2)**, we can say that

AD=CF and

**Solution (iv)**

We have showed in **solution (iii)** that AD=CF and

It means that quadrilateral ACFD is a parallelogram.

**(A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.)**

**Solution (v)**

We have showed in **solution (iv)** that ACFD is parallelogram which means that AC=DF.

**(Opposite sides of parallelogram are equal)**

**Solution (vi)**

In and

AB=DE **(Given)**

BC=EF **(Given)**

AC=DF ** (We proved above that ACFD is a parallelogram)**

Therefore, **by SSS congruence rule**,