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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 @
D ROQ
\Ð OPS @
Ð ORQ
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
çç QR.
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.
Figure 3.19
Since seg.LM çç
seg.ON
Ð LMO @
Ð NOM ( alternate angles
)
In D
LMO and D NOM
seg.LM @ seg.NO
Ð LMO @
Ð NOM
seg.MO @ seg.OM (Same
side )
\ D LMO @
D NOM
\ 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
seg.MN. Therefore 
LMNO is a parallelograme.
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