Parallelogram
A quadrilateral has both pairs of opposite sides parallel is called a parallelogram. In figure 7-5, ABCD is a parallelogram in which AB parallel to DC and AD parallel to BC . A parallelogram is denoted by the symbol ‘ ‘.
Properties of a parallelogram Property 1: A diagonal of a parallelogram divides it into two congruent triangles. Given: ABCD is a parallelogram To Prove: Triangle ABC Triangle ADC Construction: Join AC
Proof: In Triangle ABC and Triangle ACD , we have
BAC ACD ACB CAD
(Alternate interior angles) (Alternate interior angles) (Common to both the triangles) (ASA criterion of congruency)
AC=AC
Therefore Triangle ABC Triangle ADC
Hence, diagonal AC divides the parallelogram into two congruent triangles. Property 2: The opposite sides of a parallelogram are equal. Given: A parallelogram ABCD. To prove: AB=CD and DA=BC Construction: Join AC
Proof: Since ABCD is AB parallel to DC and DA parallel to BC .
a
parallelogram,
therefore,
Now, AD parallel to BC and transversal AC intersects them at A and C respectively.
Therefore DAC BCA
(Alternate interior angles)
… (1)
Now, AB DC and transversal AC intersects them at A and C respectively.
Therefore BAC DCA
(Alternate interior angles)
… (2)
Now, in ADC and CBA, we have
DAC BCA
[From (1)] (Common side to both the triangles) [From (1)]
AC=AC
BAC DCA
Therefore By ASA criterion of congruence
Triangle ADC Triangle CBA
Thus, AD=BC and DC=BA (Corresponding parts of congruent triangles are congruent).
Property 3: Opposite angles of a parallelogram are equal. Given: ABCD is a parallelogram To prove: A C and B D
Proof: In parallelogram ABCD, AB parallel to DC and DA intersects them. therefore A D 180o (Consecutive interior angles) … (1)
Again, in parallelogram ABCD, AD parallel to BC and AB intersects them Therefore A B 180o From (1) and (2), we get
A D A B
D B B D
(Consecutive interior angles)
… (2)
Similarly A C Hence A C and D B
Property 4: The diagonals of a parallelogram bisect each other. Given: A parallelogram ABCD in which the diagonals AC and AD intersect at O. To prove: OA=OC and OB=OB
Proof: Since ABCD is a parallelogram
Therefore AB parallel to DC
and
Therefore AD parallel to BC
Now, AB parallel to DC and AC is the transversal intersecting them at A and C respectively.
Therefore BAC DCA
(Alternate interior angles) … (1)
Thus, BAO DCO
As AB parallel to DC and BD intersects them at B and D respectively.
Therefore ABD CDB
(Alternate interior angles) … (2)
So, ABD CDO
Now, in Triangle AOB and Triangle COD , we have
BAO DCO
[From (1)]
AB=CD
ABO CDO
(Opposite sides of a parallelogram) [From (2)]
therefore ASA congruence criterion
Triangle AOB Triangle COD
Thus, OA=OC and OB=OB (Corresponding parts of congruent triangles are congruent) Therefore The diagonals of a parallelogram bisect each other. Property 5: The consecutive angles of a parallelogram are supplementary. Given: Parallelogram ABCD To prove: A ABC 180o Construction: Produce AB to P Proof: Since AD parallel to BC and AB is the transversal,
Therefore A CBP
(Corresponding angles) … 92)
… (1)
Now, ABC CBP 180o (They form a linear pair)
From (1) and (2), we have
ABC A 180o
( Since CBP A )
or A ABC 180o So, consecutive angles of a parallelogram are supplementary Sufficient conditions for a quadrilateral to be a parallelogram The condition, which a quadrilateral must satisfy to become a parallelogram are know as ‘sufficient conditions’. Condition 1: A quadrilateral is a parallelogram if its opposite sides are equal.
Given: A quadrilateral ABCD in which AB=CD and BC=AD To prove: ABCD is a parallelogram Construction: Join AC Proof: In Triangle ABC and Triangle ADC , AB=CD BC=AD AC=AC (Given) (Given) (Common to both the triangles)
Therefore Triangle ABC Triangle ADC (SSS criterion of congruency)
Thus, BAC DCA congruent)
(Corresponding parts of congruent triangles are
Therefore BAC and DCA are alternate interior angles.
Hence AC DC Now, BAC DCA (cpctc)
Thus, these are alternate interior angles,
Therefore BC parallel to AD
Hence, quadrilateral ABCD is a parallelogram. Condition 2: A quadrilateral is a parallelogram if its opposite angles are equal. Given: A quadrilateral ABCD, in which A C and B D To prove: Quadrilateral ABCD is a parallelogram
Proof: In quadrilateral ABCD
A C
(Given) (Given) … (1)
B D
Therefore A B C D
But A B C D 360o (Sum of the angles of a quadrilateral = 360o ) From (1) and (2), we have
A B C D 360o A B A B 360o 2A 2B 360o
… (2)
[ Since A B C D ]
2(A B) 360o 360o (A B) 2 (A B) 180o Therefore C D 180o … (3) [ Since A B C D ]… (4)
Since, transversal AB intersects lines AD and BC at A and B respectively, and (A B) 180o
Therefore A and B form a pair of consecutive interior angles and consecutive interior angles are supplementary)
So, AD parallele to BC Again, A C Therefore By using this relation in (3), we have
C B 180o
… (5)
Now, AB and DC are the lines and BC is the transversal,
Therefore B and C form a pair of consecutive interior angles and consecutive interior angles are supplementary)
Thus, AB parallel to DC From (5) an (6), we find that
AD parallel to BC and AB parallel to DC
… (6)
Hence, quadrilateral ABCD is a parallelogram Condition 3: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram Given: A quadrilateral ABCD, diagonals AC and BD intersect at O such that AO=OC and BO=OB. To prove: Quadrilateral ABCD is a parallelogram.
Proof: In Triangle AOD and Triangle COB, we have AO=OC OD=OB (Given) (Given)
AOD COB
(Vertically opposite angles are equal)
Therefore By SAS criterion of congruence,
Triangle AOD Triangle COB
So, OAD OCB
… (1) (cpctc)
Now, lines AC intersects AD and BC at A and C respectively, such that
OAD OCB
… [From (10]
Therefore OAD and OCB form a pair of alternate interior angles
(Therefore Alternate interior angles are equal) Thus, AD parallel to BC Similarly, we can prove that AB parallel to DC Hence quadrilateral ABCD is a parallelogram Condition 4: A quadrilateral is a parallelogram, if its one pair of opposite sides are equal and parallel. Given: A quadrilateral ABCD in which AB=CD and AB parallel to CD . To prove: ABCD is a parallelogram
Construction: Join AC. Proof: In Triangle ABC and Triangle CDA , we have AB=CD AC=AC
BAC DCA are equal)
(Given) (Common to both triangles)
(Since AB parallel to DC and alternate interior angles
Therefore Triangle ABC Triangle DCA
(SAS criterion of congruency)
Thus ACB CAD
(cpctc)
But these are alternate interior angles,
Therefore AD Parallel to BC
Thus we have AD parallel to BC (proved above) and AD CD (given). Hence ABCD is a parallelogram. Condition 5: A quadrilateral is a parallelogram if its opposite sides are parallel. Proof: From the definition of a parallelogram, a quadrilateral whose opposite sides are parallel is called a parallelogram, the condition holds.