Conditions for a parallelogram
The converse of the above theorems are proved below. These theorems give the conditions under which a quadrilateral is a parallelogram.
Theorem: A quadrilateral is a parallelogram,
if its opposite sides are congruent. Figure 3.17 shows a quadrilateral
ABCD with its opposite sides congruent.
To prove that ABCD is a parallelogram, join A to C and consider
D ADC &
CB and DC @
BA ( given ) also AC @ CA ( same
CBA ( SSS ).
ACB @ Ð
CAD and Ð ACD
CAB ( corresponding angles of congruent triangles are congruent
Ð ACB @ Ð CAD Þ AD çç BC because they are alternate angles formed by the transversal CD that intersects BC and AD. Since they are congruent the two lines intersected by the transversal are parallel.
Similarly it can be show that since Ð
ACD @ Ð
CAB AB çç
DC. Since both the opposite sides are parallel to each other
ABCD is a parallelogram.
Theorem: A quadrilateral is a parallelogram
if itís diagonals bisect each other. Figure 3.18 shows a quadrilateral
PQRS such that its diagonals seg.PR and seg.QS bisect each other
on the point of intersection O.