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Thus it is proved that the opposite sides of a parallelogram are
congruent. From the same proof it can be said that the diagonal
of a parallelogram divides it into two congruent triangles.
Since D
ACB @
D CAD
Ð ABC
@ Ð
CDA
By drawing a diagonal from D to B it can be shown
that Ð DAB
@ Ð
BCD which means that in a parallelogram the opposite angles
are congruent.
Another important feature of a parallelogram is given in the theorem below :
Theorem: The diagonals of a parallelogram
bisect each other. Figure 3.16 shows a parallelogram PQRS, seg.PR
and seg.QS are its two diagonals that intersect in O.
Figure 3.16
To prove that seg.PR & seg.QS bisect each other at O.
In D
SOR and D
QOP, Ð
OSR @ Ð
OQP and Ð
ORS @ Ð
OPQ (alternate angles ).
SR @ PQ opposite sides
of a parallelogram.
\ D
SOR @ D
QOP ( ASA )
\ seg. SO @
seg.OQ i.e. O is the midpoint of SQ and seg.PO @
seg.OR i.e. O is the midpoint of PR. Hence it is proved that PR
and SQ bisect each other at O.
Summary of the properties of a Parallelogram
1) Both the pairs of opposite sides of a parallelogram are parallel to each other.
2) The opposite sides of a parallelogram are congruent.
3) The opposite angles of a parallelogram are congruent.
4) The two triangles, formed by a diagonal of a parallelogram, are congruent.
5) The diagonals of a parallelogram bisect each other.
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