Properties of Parallelogram


 
 
Concept Explanation
 

Properties of Parallelogram

PROPERTIES OF PARALLELOGRAM:

A quadrilateral is a prallelogram if its both pairs of opposite sides are parallel to each other. In fig., quadrilateral ABCD is a prallelogram, because AB large parallel DC and AD large parallel BC. Now let us discuss some theorems

Theorem 1: A diagonal of a prallelogram divides it into two congruent triangles.

Theorem 2: In parallelogram, opposite sides are equal.

Theorem 3: The opposite angles of a parallelogram are equal.

Theorem 4: The diagonals of a parallelogram bisect each other.

Theorem 5: In a parallelogram, the bisectors of any two consecutive angles intersect at right angle.

Theorem 6: If diagonal of a parallelogram bisects one of the angles of the parallelogram, it also bisects the second angles. Also, prove that it is a rhombus.

Theorem 7: The angle bisectors of a parallelogram form a rectangle.

ILLUSTRATION: In a parallelogram ABCD two points P and Q are taken on the diagonal BD such that DP = BQ. Show that

(1)  Delta APD cong Delta CQB                                           (2)  AP = CQ                                                 (3)   Delta AQB cong Delta CPD

(4)   AQ= CP                                                                 (5) APCQ is a parallelogram

Solution: In Delta APD ;and; Delta CQB

         AD = CB         [ opposite sides of a parallelogram]

   angle ADP = angleQBC   [ Alternate interior angles whne AD || BC and BD is the transversal]

       DP  = QB          [ Given]

   Delta APD cong Delta CQB    [ SAS criteria of congruence]    Part (1) Proved

   angletherefore  AP = CQ     [ C.P.C.T.]                                          Part (2) Proved

  In Delta AQB ;and; Delta CPD

         AB = CD         [ opposite sides of a parallelogram]

   angle ABQ = angleCDP   [ Alternate interior angles whne AB || CD and BD is the transversal]

       BQ  = DP          [ Given]

   Delta AQB cong Delta CPD    [ SAS criteria of congruence]    Part (3) Proved

   angletherefore  AQ = CP     [ C.P.C.T.]                                          Part (4) Proved

Now in the quadrilateral both the opposite pair of sides are equal.

therefore  APCQ is a parallelogram                                             Part (5) Proved 

 

Sample Questions
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Question : 1

In the figure given below AP and CQ are perpendiculars to the diagonal BD of a parallelogram. Then

Right Option : A
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Question : 2

The diagonals of a parallelogram ABCD intersect at O. If <BOC=90 and <BDC 50 ,then <OAB is :

Right Option : B
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Question : 3

ABCD is a parallelogram.The angle bisectors of <A and <D meet at O.The meausre of <AOD is:

Right Option : B
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Explanation
 
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