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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?
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