If we look around we will see quadrilaterals everywhere. The floors, the ceiling, the blackboard in your school, also the windows of your house. So along with the quadrilaterals, let us also study their properties of quadrilateral shapes in detail.

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We can define quadrilateral as a figure formed by joining four sides. There are also properties associated with a quadrilateral which we are going to study further. In a quadrilateral, one amazing aspect is that it can have parallel opposite sides. They have four sides, four vertices, and four angles. The sum of the internal angles of the quadrilateral is 360 degree.

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The main property of a quadrilateral is Angle sum Property of Quadrilateral which states that the sum of the angles of the quadrilateral is 360°.

In the above figure, we see a quadrilateral ABCD and AC is the diagonal of the quadrilateral.

∠A + ∠B + ∠C + ∠D = 360°

which means the sum of all angles of a quadrilateral is  360°.

• A trapezium is a quadrilateral wherein one pair of the opposite sides are parallel while the other isn’t.
• The rectangle is a plane shape with four sides. It is a 4 sided polygon with opposite sides parallel.
• A parallelogram is a slanted rectangle with the length of the opposite sides being equal just like a rectangle. Because of the parallel lines, opposite sides are equal and parallel.
• A rhombus is a quadrilateral which means it has four sides. So, just like a square with congruent or equal sides.

Squares, rectangles and rhombus are all parallelograms.

### Properties of a Parallelogram

• Opposite sides are congruent.
• Opposite angles are congruent.
• If one angle is right, then all angles are right.
• The diagonals of a parallelogram bisect each other.

### 1.  If the diagonals of a quadrilateral bisect each other then it is a parallelogram.

Let us discuss some properties of quadrilateral shapes.

Given: The above figure PQRS is a quadrilateral in which the diagonals PR and QS intersect in M.
PM = RM and QM = SM

To Prove: PQRS is s quadrilateral

Proof: In Δ PMQ and Δ RMS,
PM = RM and QM = SM
Also,  ∠PMQ = ∠RMS
∴ Δ PMQ ≅ Δ RMS
∴ ∠PQM = ∠RSM
or, PQ || SR

Similarly, we can prove that PS || QR or we can say PQRS is a parallelogram. Hence, if the diagonals of a quadrilateral bisect each other then it is a parallelogram.

### 2. If a pair of opposite side of a quadrilateral is parallel and congruent then the quadrilateral is a parallelogram.

Given: LMNK is a given quadrilateral, LM||NK and LM = NK

To prove: LMNK is a parallelogram

Proof: LMNK is a given quadrilateral in which side LM|| side NK and MK is transversal.
∠LMK = ∠NKM
Now in ΔKLM and ΔMNK
LM = NK
∠LMK = ∠NKM
KM = MK
∴ ΔKLM = ΔMNK
∴∠LKM = ∠NMK
∴ LK || MN and LM||NK
So LMNK is a parallelogram. Hence if a pair of opposite side of a quadrilateral is parallel and congruent then the quadrilateral is a parallelogram.

### 3. The diagonals of the parallelogram bisect each other.

Moving on to another of the Properties of Quadrilaterals

Given: ABCD is a parallelogram in which the diagonals AC and BD intersect in M.
seg AM = seg CM and seg BM  = seg DM

Proof: Since ABCD is a parallelogram
∠BAC  = ∠DCA
∠BAM  = ∠DCM
Also, side AB || side DC
∴ ∠ABD  = ∠CDB
∠ABM  = ∠CDM
Now, In ΔABM and ΔCDM
∠BAM  = ∠DCM
AB ≅ DC
∠ABM  = ∠CDM
∴ Δ ABM ≅ ΔCDM
∴ seg AM = seg CM and seg BM = seg DM
Hence, The diagonals of the parallelogram bisect each other.

### 4.  The opposite angles of a parallelogram are congruent.

Given: In ABCD diagonal AC ⊥ BD
seg AM = seg CM and seg DM = seg BM

To prove: ABCD is a rhombus

Proof: In Δ AMB and ΔCMB,
seg AM  = seg CM
∠AMB = ∠CMB
seg MB = seg MB
∴ Δ AMB ≅ Δ CMB
∴ seg AB = seg CB
Similarly, we can prove that, ΔCMB ≅ ΔCMD
∴ seg AB = seg CD
We can also prove that, ΔCMD ≅Δ AMD
∴ seg CD = seg AD
Thus, seg AB = seg CB = seg CD = seg AD
Hence, If the diagonals of a quadrilateral bisect each other at the right angle then it is a rhombus.

### 5. If diagonals of a parallelogram are congruent then it is a rectangle.

Given: ABCD is a parallelogram in which diagonal AC ≅ DB

To prove: ABCD is a rectangle

Proof: In ΔABC and Δ DCB,
seg AC = seg DB
seg AB = seg DC
Also, seg BC = seg CB
∴ Δ ABC ≅ Δ DCB
∴ ∠ ABC = ∠DCB
Also, ∠ ABC and  ∠DCB are supplementary
∴ ∠ ABC = ∠DCB = 90°
We can also prove that ∠BAD = ∠CDA = 90°
∴ ABCD is a rectangle. Hence, If diagonals of a parallelogram are congruent then it is a rectangle.

## Solved Question

Q. If ABCD is a parallelogram with two adjacent angles A and B equal to each other, then the parallelogram is?

1. Rhombus
2. Trapezium
3. Rectangle
4. None of these

Solution: C. According to one of the Properties of Quadrilaterals, the sum of any two sides of the parallelogram is 180°
∠A + ∠ B = 180°
∠A = ∠ B = 90°
Similarly ∠C = ∠ D = 90°
Since all angles are 90° it is a rectangle.

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