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 = $angle$ QBC [ 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 = $angle$ CDP [ 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

(More Questions for each concept available in Login)

Question : 1

ABCD is a parallelogram.The angle bisectors of <A and <D meet at O.The meausre of <AOD is:
A. 45
B. 90
C. dependent on the angles A and D
D. cannot be determined from given data
Right Option : B
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Question : 2

The perimeter of a parallelogram is 180 cm. One side exceeds the other by 30 cm. The adjacent sides of the parallelogram are ____________
A. 30 cm, 40 cm	B. 60 cm, 30 cm	C. 50 cm, 60 cm	D. 20 cm, 50 cm
Right Option : B
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Question : 3

ABCD is a quadrilateral.If AC and BD bisect each other then ABCD must be:
A. square
B. rectangle
C. parallelogram
D. rhombus
Right Option : C
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