Quadrilateral is a Parallelogram

Key Concepts

• Prove that a quadrilateral is a parallelogram.
• Find the relation between the angles of a parallelogram.
• Use properties of sides, and diagonals to identify a parallelogram.

Parallelogram 

A simple quadrilateral in which the opposite sides are of equal length and parallel is called a parallelogram. 

Real-life examples of a parallelogram 

1. Dockland office building in Hamburg, Germany. 

2. An eraser 

3. A striped pole 

4. A solar panel 

Theorem 

If one pair of opposite sides of a quadrilateral is parallel and congruent, then the quadrilateral is a parallelogram. 

Given: WX=ZY

To prove: WXYZ is a parallelogram. 

Proof:  Now, in △XZY and △ZXW,

WX = ZY

∠WXZ = ∠XZY [Alternate interior angles]

XZ = ZX [Reflexive property] 

So, △XZY ≅ △ZX by Side-Angle-Side congruence criterion 

If two triangles are congruent, their corresponding sides are equal. 

Hence, WZ=XY

Therefore,

WXYZ is a parallelogram. 

Theorem 

If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. 

Given: ∠A+∠B = 180° and ∠A+∠D = 180°

To prove: ABCD is a parallelogram. 

Proof: 

Given: ∠A and ∠B are supplementary. 

If two lines are cut by a transversal, then the consecutive interior angles are supplementary, and the lines are parallel. 

So, AD ∥ BC …(1) 

Given: ∠A and ∠D are supplementary. 

We know that if the consecutive interior angles are supplementary, then the lines are parallel. 

So, AB ∥ CD …(2) 

From (1) and (2), we get AB∥CD and AD∥BC

Hence,

ABCD is a parallelogram. 

Theorem 

If both the pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. 

Given: PQ=RS and PS=QR

To prove: PQRS is a parallelogram. 

Proof:  

Now, in △PQS and △RSQ,

PQ = RS

PS = RQ

QS = SQ [Reflexive property] 

So, △PQS ≅  △RSQ by Side-Side-Side congruence criterion 

If two triangles are congruent, their corresponding angles are equal. 

Hence, ∠PQS = ∠RSQ and ∠QSP = ∠SQR

We know that if two lines are cut by a transversal, then the alternate interior angles are congruent, and the lines are parallel. 

So, PQ ∥ SR and PS ∥ QR

Therefore,

PQRS is a parallelogram. 

Theorem 

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. 

Given: LP=PN and OP=PM 

To prove: LMNO is a parallelogram. 

Proof:  

Now, in

△OPN and △MPL, OP=MP

∠OPN=∠MPL [Vertically opposite angles] 

PN=PL

So,

△OPN ≅  △MPL by Side-Angle-Side congruence criterion. 

If two triangles are congruent, their corresponding sides are equal. 

Hence,

ON = LM  …(1) 

And, in △LPO and △NPM,

LP = NP

∠LPO = ∠NPM [Vertically opposite angles] 

PO = PM

So, △LPO ≅  △NPM by Side-Angle-Side congruence criterion. 

If two triangles are congruent, their corresponding sides are equal. 

Hence,

LO = MN …(2) 

From (1) and (2), we get

ON = LM and LO = MN. 

∴LMNO is a parallelogram. 

Theorem 

If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. 

Given: ∠A=∠C; ∠B=∠D

To prove: ABCD is a parallelogram. 

Proof:  

We know that the sum of angles of a quadrilateral is 360°. 

So,

∠A+∠B+∠C+∠D=360°

∠A+∠B+∠A+∠B=360°

2(∠A+∠B) =360°

∠A+∠B=180°

If two lines are cut by a transversal, then the consecutive interior angles are supplementary, and the lines are parallel. 

Since ∠A and ∠B are supplementary,

so, AD ∥ BC …(1) 

Or we can write 

∠A+∠B+∠C+∠D=360°

∠A+∠D+∠A+∠D=360°

2(∠A+∠D) =360°

∠A+∠D=180°

Since the angles ∠A and ∠D are supplementary, so,

AB ∥ CD       …(2) 

From (1) and (2), we get AB ∥ CD and AD ∥ BC

So, ABCD is a parallelogram. 

Exercise

• For what values of x and y is the given quadrilateral a parallelogram?

• For what values of w and z is the given figure a parallelogram?

• Is the figure below a parallelogram?

• For what values of x and y is the given quadrilateral a parallelogram?

• Is the figure below a parallelogram?

Concept Map

What we have learned

• A quadrilateral whose two pairs of opposite sides are parallel is called a parallelogram.
• If a quadrilateral is a parallelogram, then its opposite sides are congruent.
• If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
• If a quadrilateral is a parallelogram, then opposite angles are congruent.
• If a quadrilateral is a parallelogram, then its diagonals bisect each other.