## 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

**Dockland office building in Hamburg, Germany.**

**An eraser**

**A striped pole**

**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.

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