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,
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