In view of the coronavirus pandemic, we are making LIVE CLASSES and VIDEO CLASSES completely FREE to prevent interruption in studies
Maths > Quadrilaterals > Properties of Quadrilaterals
Quadrilaterals

Properties of Quadrilaterals

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.

Suggested Videos

Play
Play
Play
Arrow
Arrow
ArrowArrow
Proof of angle sum property of quadrilaterals
Applications of angle sum property of a quadrilateral
Prove diagonals of rhombus are perpendicular bisector of each other
Slider

 

Quadrilateral Shapes

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.

Browse more Topics under Quadrilaterals

Properties of Quadrilateral Shapes

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°.

Properties of Quadrilaterals

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°.

Types Of  Quadrilateral Shapes

Properties of Quadrilaterals

  • 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.

Theorems of Quadrilateral Shapes

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

Properties of Quadrilaterals

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.

Properties of Quadrilaterals

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.

Quadrilateral Shapes

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.

Quadrilateral Shapes

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.

Quadrilateral Shapes

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.

Share with friends

Customize your course in 30 seconds

Which class are you in?
5th
6th
7th
8th
9th
10th
11th
12th
Get ready for all-new Live Classes!
Now learn Live with India's best teachers. Join courses with the best schedule and enjoy fun and interactive classes.
tutor
tutor
Ashhar Firdausi
IIT Roorkee
Biology
tutor
tutor
Dr. Nazma Shaik
VTU
Chemistry
tutor
tutor
Gaurav Tiwari
APJAKTU
Physics
Get Started

Leave a Reply

avatar
  Subscribe  
Notify of

Stuck with a

Question Mark?

Have a doubt at 3 am? Our experts are available 24x7. Connect with a tutor instantly and get your concepts cleared in less than 3 steps.
toppr Code

chance to win a

study tour
to ISRO

Download the App

Watch lectures, practise questions and take tests on the go.

Get Question Papers of Last 10 Years

Which class are you in?
No thanks.