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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