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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.
A quadrilateral ABCD has
There are many kinds of quadrilaterals. Such as:
Quadrilaterals having opposite sides parallel is known as a parallelogram.
In the figure AB ⁄⁄ CD and AD ⁄⁄ BC. So, ABCD is a parallelogram.
Theorems with parallelogram:
The opposite sides of a parallelogram are congruent.
Verification:
Draw three parallelograms of different sizes as shown below:
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.
The opposite angles of a parallelogram are congruent.
Verification:
Draw three parallelograms of different sizes.
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.
The diagonals of a parallelogram bisect each other.
Verification:
Draw three parallelograms of different sizes. Join the diagonals WY and XZ.
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.
The rectangle is a parallelogram with all angles 90^{o.} Opposite sides are parallel and of equal length. It is also known as an equiangular parallelogram.
Diagonal created in a rectangle are also congruent.
The diagonals of a rectangle are congruent.
Verification:
Draw three rectangles of different sizes. Join the diagonals WY and XZ.
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.
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.
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.
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.
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=70^{o}
\(\angle\)DAB=x=?
Now,
\(\angle\)DAB+\(\angle\)ADC=180^{o }[sum of co-interior angles of parallelogram is 180]
or, x+70^{o}=180^{o}
or, x=180^{o}-70^{o}
\(\therefore\) x =110^{o}
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 x^{o} and y^{o}.
We know that, the diagonals of a square bisects each other perpendicularly.
So, x= y=90^{o}
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 = 90^{o}
\(\angle\)HEF = x^{o}
\(\angle\)EFG = y^{o}
\(\angle\)FGH = z^{o}
Now,
\(\angle\) HGF = \(\angle\)EFG = \(\angle\)FGH = 90^{o } [Angles of rectangle are equal]
\(\therefore\) x = 90^{o}, y = 90^{o} and z = 90^{o}
Solution:
Here, \(\angle\) A=\(\angle\)B=\(\angle\)C=\(angle\)=90^{o}
So, All angle are equal and are 90^{o}of 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:
(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.
Find the value of x, from the following figure.
Find the value of x, from the following figure?
ASK ANY QUESTION ON Quadrilaterals
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