Parallelogram is a quadrilateral whose opposite sides are parallel and congruent. Their opposite angles are equal and consecutive angles are supplementary. The diagonals bisect each other dividing the parallelogram into congruent triangles.

## Parallelogram Theorem Proofs

### The theorems of parallelogram are as follows:

i) In a parallelogram opposite sides are congruent.

ii) In a parallelogram opposite angles are equal.

iii) In a parallelogram consecutive angles are supplementary.

iv) In a parallelogram, diagonals bisect each other.

v) The diagonal divides the parallelogram into two congruent triangles.

Proof: In a parallelogram opposite sides are congruent

Given: ABCD is a parallelogram
AB//CD

STATEMENT REASON
Angle ADB=Angle CBD Alternate Interior Angles
Angle ABD=Angle BDC Alternate Interior Angles
DB congruent DB Reflexive (Identity)
Triangle ADB congruent Triangle CBD ASA
AB congruent CD CPCTC
BC congruent DA CPCTC

Therefore, in a parallelogram ABCD opposite sides AB, CD and BC, DA are congruent to each other.

Proof: In a parallelogram, opposite angles are equal.

Given: ABCD is a parallelogram
AB//CD

Let us draw a line BD joining two points B and D

STATEMENT REASON
Angle ADB=Angle CBD Alternate Interior Angles
Angle ABD=Angle BDC Alternate Interior Angles
DB = DB Reflexive (Identity)
Triangle ADB $\cong$ Triangle CBD ASA
$\angle$ DAB = $\angle$ DCB CPCTC
$\angle$ ADB = $\angle$ CBD$\angle$ ADB + $\angle$ CDB Corresponding angles have equal measure
$\angle$ ADB + $\angle$ CDB =$\angle$ ABD + $\angle$ CBD Addition of equalities
$\angle$ ABD + $\angle$ CBD $\angle$ ADB + $\angle$ CDB = $\angle$ ADC $\angle$ ABD + $\angle$ CBD =Measure of Angle ABC Angle addition postulate
$\angle$ ADC= $\angle$ ABC Substitution
$\angle$ ADC = $\angle$ ABC Congruent angles have equal measures

Therefore, in a parallelogram opposite angles are equal.

Proof: In a parallelogram consecutive angles are supplementary

Given: ABCD is a parallelogram
AB//CD

Therefore, angle A and angle B are supplementary, angle B and angle C are supplementary, angle C and angle D are supplementary and angle D and angle A are supplementary angles. Reason being if two parallel lines are cut by a transversal, then the interior angles lying on the same side are supplementary angles.

Proof: In a parallelogram, diagonals bisect each other.

Given: ABCD is a parallelogram.

Let diagonals AC and BD intersect at a point O.

Statement Reasons
AB//CD Definition of parallelogram
Angle BAO = Angle DCO Alternate Interior Angles
AB = CD Opposite sides of parallelogram
Angle ABO = Angle CDO Alternate Interior Angles
Triangle ABO $\cong$ Triangle CDO ASA postulate
AO = OC CPCTC
BO = OD CPCTC

Therefore, in a parallelogram, diagonals bisect each other.

Proof: The diagonal divides the parallelogram into two congruent triangles

Given: ABCD is a parallelogram
AB//CD

$\left.\begin{matrix} \angle 1 = \angle 6\\ \angle 2 = \angle 5\\ \angle 3 = \angle 7\\ \angle 4 = \angle 8\\\end{matrix}\right\}$
If two parallel lines are cut by a transversal then the alternate interior angles are congruent

$\left.\begin{matrix}AC = AC\\ DB = DB\end{matrix}\right\}$ Reflexive property

$\left.\begin{matrix} Triangle\ ADC\ \cong\ Triangle\ CBA\\ Triangle\ ADB\ \cong\ Triangle\ CBD \end{matrix}\right\}$ ASA postulate of congruency.

Therefore, diagonal divides the parallelogram into two congruent triangles.

## Parallelogram Area Theorem

Area of parallelogram is the product of base and height.

Theorem of parallelogram area is as follows:

Parallelograms on the same base and between the same parallels are equal in area.

Proof: Parallelograms on the same base and between the same parallels are equal in area

Let us consider two parallelograms ABCD and EFCD

Angle DAE = Angle CBF (AD//BC and transversal AF passing making Angle DAE and CBF corresponding angles)

Angle DEA = Angle CFB (CF//DE and transversal AF passing making Angle DEA and CFB corresponding angles)

Therefore, Angle ADE = Angle BCF (Sum of angles in a triangle is 180 degrees)

Also, AD = BC (Property of parallelogram AD and BC opposite sides of parallelogram ABCD equal)

So, Triangle ADE congruent Triangle BCF (ASA postulate of congruency)

Therefore, area of triangle ADE=Area of triangle BCF

Area of ABCD = Area of triangle ADE + Area of EDCB
= Area of triangle BCF + Area of EDCB
= Area of EFCD

So, parallelogram ABCD and parallelogram EFCD are equal in area

## Parallelogram Theorem Vectors

The parallelogram law of vector addition states that if two vector quantities are represented by two adjacent sides of a parallelogram, then the diagonal represents the resultant vector.

$Vector\ R$ = $Vector V_{1}\ +\ Vector V_{2}$

## Converse of Parallelogram Theorems

i) If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.

AB = CD

Let us consider triangles ABD and BCD

Statement Reasons
AB=CD Given
BD=BD Common side to both the triangles

Therefore, triangle ABD congruent triangle BCD (SSS postulate of congruency)

So, Angle ABD = Angle CDB (By CPCTC)
Angle ADB = Angle CBD (By CPCTC)

Hence, Quadrilateral ABCD is a parallelogram (Proved)

ii) If in a quadrilateral each pair of opposite angles is equal, then it is a parallelogram.

Angle A = Angle C ........(1)

Angle B = Angle D  .......(2)

Adding equation 1 and equation 2, we get

Angle A + Angle B=Angle C + Angle D

It is known to us that sum of interior angles of a quadrilateral is 360

Angle A + Angle B + Angle C + Angle D = 360

Angle A+ Angle B+ Angle A+ Angle B = 360

2 $\times$ (Angle A + Angle B) = 360

Angle A + Angle B = 180

Similarly, Angle A + Angle D = 180

AB//CD

Hence, quadrilateral ABCD is a parallelogram as opposite sides are parallel (Proved)

iii) Proof: If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

Let us consider triangles AOB and COD,

Given:
AO = OC
DO = OB

Angle AOB = Angle DOC (Vertically opposite angles)

Triangle AOB $\cong$ Triangle COD (SAS postulate of congruency)

Angle ABO = Angle ODC (CPCTC)

Angle ABD = Angle BDC (CPCTC)

As they are alternate interior angles made by the transversal BD intersecting AB and CD, we can say AB//CD.

Similarly let us consider the other two triangles AOD and BOC,

Given:
AO = OC
DO = OB

Angle AOD = Angle BOC (Vertically opposite angles)

Therefore, triangle AOD congruent Triangle BOC (SAS postulate of congruency)

Angle OAD = Angle OCD (CPCTC)
Angle ODA = Angle OBC (CPCTC)

As they are alternate interior angles made by the transversal AC intersecting AD and BC, we can say AD//BC

So, we can conclude that if the diagonals of a quadrilateral bisect each other making the opposite sides parallel to each other, then it is a parallelogram.