#### Need Help?

Get in touch with us

## 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.
1. An eraser
1. A striped pole
1. 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.

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,

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?

### 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.

#### Composite Figures – Area and Volume

A composite figure is made up of simple geometric shapes. It is a 2-dimensional figure of basic two-dimensional shapes such as squares, triangles, rectangles, circles, etc. There are various shapes whose areas are different from one another. Everything has an area they occupy, from the laptop to your book. To understand the dynamics of composite […] #### Special Right Triangles: Types, Formulas, with Solved Examples.

Learn all about special right triangles- their types, formulas, and examples explained in detail for a better understanding. What are the shortcut ratios for the side lengths of special right triangles 30 60 90 and 45 45 90? How are these ratios related to the Pythagorean theorem?  Right Angle Triangles A triangle with a ninety-degree […] #### Ways to Simplify Algebraic Expressions

Simplify algebraic expressions in Mathematics is a collection of various numeric expressions that multiple philosophers and historians have brought down. Talking of algebra, this branch of mathematics deals with the oldest concepts of mathematical sciences, geometry, and number theory. It is one of the earliest branches in the history of mathematics. The study of mathematical […]   