Note on Quadrilaterals

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A closed plane figure formed by four line segments is a quadrilateral. A quadrilateral is also known as a polygon with four sides and four vertices or corners.

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A quadrilateral ABCD has

  • four sides: AB, BC, CD and DA
  • four angles:∠A, ∠B,∠C and ∠D

There are many kinds of quadrilaterals. Such as: 

1. Parallelogram

Quadrilaterals having opposite sides parallel is known as a parallelogram.

In the figure AB ⁄⁄ CD and AD ⁄⁄ BC. So, ABCD is a parallelogram.

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Theorems with parallelogram:

Theorem 1

The opposite sides of a parallelogram are congruent.

Verification:

Draw three parallelograms of different sizes as shown below:

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Measure the sides and complete the table below:

Figure WZ XY Result WX ZY Result
(i)     WZ=XY     WX = ZY
(ii)            
(iii)            

Conclusion: Opposite sides of a parallelogram are equal.

Theorem 2

The opposite angles of a parallelogram are congruent.

Verification:

Draw three parallelograms of different sizes.

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Measure the opposite angles and complete the table below:

Figure ∠W ∠Y Result ∠X ∠Z Result
(i)     ∠W =∠Y     ∠X =∠Z
(ii)            
(iii)            

Conclusion: The opposite angles of a parallelogram are congruent.

Theorem 3

The diagonals of a parallelogram bisect each other.

Verification:

Draw three parallelograms of different sizes. Join the diagonals WY and XZ.

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Measure the segments of the diagonals and complete the table below:

Figure WO YO Result XO ZO Result
(i)     WO = YO     XO =ZO
(ii)            
(iii)            

Conclusion: Diagonals of the parallelogram bisect each other.

 2. Rectangle

The rectangle is a parallelogram with all angles 90o. Opposite sides are parallel and of equal length. It is also known as an equiangular parallelogram.

Diagonal created in a rectangle are also congruent.

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Theorem

The diagonals of a rectangle are congruent.

Verification:

Draw three rectangles of different sizes. Join the diagonals WY and XZ.

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Measure the diagonals WY and XZ with the ruler and complete the following table.

Figure WX XZ Result
(i)     WX = XZ
(ii)      
(iii)      

Conclusion: The diagonals of the rectangle are congruent.

 3. Square

Square is also a parallelogram with all sides and angles equal. It is also known as an equilateral and equiangular parallelogram. In another word, a square is a rectangle having adjacent sides equal. The diagonal of square bisects each other at right angles.

Theorem

The diagonals of a square bisect each other at right angles.

Verification:

Draw three squares of different sizes. Join the diagonals WY and XZ which intersect at O. Since a square is a parallelogram, the diagonals bisect each other i.e WO =YO and XO = ZO.

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Measure the angles between the diagonals and complete the following table.

Figure ∠WOX ∠YOZ ∠WOZ ∠XOY Result
(i)         ∠WOX =∠YOZ =∠WOZ =∠XOY = 90°
(ii)          
(iii)          

Conclusion: The diagonals of a square bisect each other at right angles.

  • Quadrilateral just means 'four sides' (quad means four, lateral means side).
  • Quadrilateral are simple (not self-intersecting) or complex (self- intersecting) and also called crossed.
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Very Short Questions

Solution:

Given,

QS = 15cm

TQ = SR

QR = TS

Now,

TR=QS =15cm, [diagonals of a rectangle are equal]

Again,

QP=\(\frac{1}{2}\) QS [half of diagonal]

=\(\frac{1}{2}\) \(\times\) 15cm [half of diagonals]

=7.5 cm

Also,

QP =PS = 7.5cm [Half of daigonal are equal]

And,

TP=\(\frac{1}{2}\) TR [half of diagonal]

=\(\frac{1}{2}\) \(\times\) 15cm [half of diagonals]

=7.5 cm

Also,

TP = PR = 7.5cm [Half of daigonal are equal]

\(\therefore\) TP = PR = QP = PS = 7.5cm

solution:

ABCD is a parallelogram.

Here,\(\angle\)ADC=70o

\(\angle\)DAB=x=?

Now,

\(\angle\)DAB+\(\angle\)ADC=180o [sum of co-interior angles of parallelogram is 180]

or, x+70o=180o

or, x=180o-70o

\(\therefore\) x =110o

Solution:

PQRS is a square which has diagonal QS=5 cm ,

We know that, diagonal of a square are equal. So, QS=PR

\(\therefore\) PR = 5 cm

Solution:

PQRS is a square in which diagonal are PR and QS. Angle of diagonal xo and yo.

We know that, the diagonals of a square bisects each other perpendicularly.

So, x= y=90o

Solution:

Given,

AB = 8cm, AD = 6cm, CD = xcm and BC = ycm

Now,

AB = CD and AD = BC [Oposite sides of rectangle are equal]

\(\therefore\) x = AB = 8cm and y = AD = 6cm

Solution:

Here,

Given,

\(\angle\)EHG = 90o

\(\angle\)HEF = xo

\(\angle\)EFG = yo

\(\angle\)FGH = zo

Now,

\(\angle\) HGF = \(\angle\)EFG = \(\angle\)FGH = 90o [Angles of rectangle are equal]

\(\therefore\) x = 90o, y = 90o and z = 90o

Solution:

Here, \(\angle\) A=\(\angle\)B=\(\angle\)C=\(angle\)=90o

So, All angle are equal and are 90oof rectangle.

Diagonal AC=BD=20.2 cm

So, Diagonal are equal in rectangle

AD=DC=10.1 cm and BO=CO=10 cm

so, diagonal of a rectangle bisects each other.

AB=CD=18 cm and BC=AD=9 cm

so, Opposite side of a rectangle is equal

 

Steps of Construction:

Steps of Construction of a Parallelogram

(i) Draw AB = 5.2 cm.

(ii) With A as center and radius 3.2 cm, draw an arc.

(iii) With B as center and radius 3 cm draw another arc, cutting the previous arc at O.

(iv) Join OA and OB.

(v) Produce AO to C such that OC = AO and produce BO to D such that OD = OB.

(vi) Join AD, BC and CD.

Then, ABCD is the required parallelogram.

0%
  • Find the value of x, from the following figure.

     x= 160o
     x=105o
     x= 95o
     x= 110o
  • Find the value of x, from the following figure?

     x= QS= 5cm
     x= QS= 6cm
     x=QS= 7cm
     x= QS= 5.5cm
  • You scored /2


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