To prove that PQRS is a parallelogram, consider D
POS and D ROQ.
seg.PO @ seg.RO
and seg.OS @ seg.OQ ( given )
Ð POS @
Ð ROQ ( vertical angles )
\D POS @
\Ð OPS @
corresponding angles of congruent triangles are congruent.
These are alternate angles formed on seg.PS and
seg.QR by the transversal seg.PR and since they are congruent PS
Similarly by showing that Ð
OSR @ Ð
OQP, it can be shown that PQ çç
SR. Since both the pairs of opposite sides are parallel lines PQRS
is a parallelogram.
Theorem: If one pair of opposite sides is
parallel and congruent, the quadrilateral is a parallelogram.
In figure 3.19 there is a quadrilateral LMNO where seg.LM @
seg.NO and seg.LM@
seg.NO . To prove that LMNO is a parallelogram.
Since seg.LM çç
Ð LMO @
Ð NOM ( alternate angles
LMO and D NOM
seg.LM @ seg.NO
Ð LMO @
seg.MO @ seg.OM (Same
\ D LMO @
\ Ð LOM @
Ð NMO ( corresponding angles
of congruent triangles are congruent ).
But they are alternate angles formed by seg.MO
on seg.MN and seg.LO and since they are congruent seg.LO çç
LMNO is a parallelograme.