In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Figure). Show that:
(i)
(ii) AP = CQ
(iii)
(iv) AQ = CP
(v) APCQ is a parallelogram
Solution (i)
In
and
, we have
DP=BQ (Given)
(Alternate Interior angles,
)
AD=BC (Opposite sides of parallelogram are equal)
Therefore, by SAS congruence rule,
Solution (ii)
We have already showed in solution (i) that
.
(Corresponding parts of congruent triangles are equal)
Solution (iii)
In
and
, we have
BQ=DP (Given)
(Alternate Interior angles,
)
AB=CD (Opposite sides of parallelogram are equal)
Therefore, by SAS congruence rule,
Solution (iv)
We have showed in solution (iii) that
(Corresponding parts of congruent triangles are equal)
Solution (v)
We have showed in solution (ii) and solution (iv) that AP=CQ and AQ=CP.
It means that APCQ is a paralleogram.
(If in a quadrilateral, each pair of opposite sides is equal, then it is a parallelogram.)
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