SELINA Solutions for Class 9 Maths Chapter 14 - Rectilinear Figures [Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium]

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Chapter 14 - Rectilinear Figures [Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium] Exercise Ex. 14(A)

Question 1

The sum of the inteior angles of a polygon is four times the sum of its exterior angles. Find the number of sides in the polygon.

Solution 1

The sum of the interior angle=4 times the sum of the exterior angles.

Therefore the sum of the interior angles = 4×360° =1440°.

Now we have

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Thus the number of sides in the polygon is 10.

Question 2

The angles of a pentagon are in the ratio 4 : 8 : 6 : 4 : 5. Find each angle of the pentagon.

Solution 2

Let the angles of the pentagon are 4x, 8x, 6x, 4x and 5x.

Thus we can write

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Hence the angles of the pentagon are:

4×20Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium= 80Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium, 8×20Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium= 160Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium, 6×20Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium= 120Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium, 4×20Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium= 80Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium, 5×20Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium= 100Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Question 3

One angle of a six-sided polygon is 140o and the other angles are equal. Find the measure of each equal angle.

Solution 3

Let the measure of each equal angles are x.

Then we can write

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Therefore the measure of each equal angles are 116Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Question 4

In a polygon there are 5 right angles and the remaining angles are equal to 195o each. Find the number of sides in the polygon.

Solution 4

Let the number of sides of the polygon is n and there are k angles with measure 195o.

Therefore we can write:

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

In this linear equation n and k must be integer. Therefore to satisfy this equation the minimum value of k must be 6 to get n as integer.

Hence the number of sides are: 5 + 6 = 11.

Question 5

Three angles of a seven sided polygon are 132o each and the remaining four angles are equal. Find the value of each equal angle.

Solution 5

Let the measure of each equal angles are x.

Then we can write:

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Thus the measure of each equal angles are 126o.

Question 6

Two angles of an eight sided polygon are 142o and 176o. If the remaining angles are equal to each other; find the magnitude of each of the equal angles.

Solution 6

Let the measure of each equal sides of the polygon is x.

Then we can write:

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Thus the measure of each equal angles are 127o.

Question 7

In a pentagon ABCDE, AB is parallel to DC and Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumA : Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumE : Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumD = 3 : 4 : 5. Find angle E.

Solution 7

Let the measure of the angles are 3x, 4x and 5x.

Thus

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Thus the measure of angle E will be 4×30Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium=120Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Question 8

AB, BC and CD are the three consecutive sides of a regular polygon. If Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumBAC = 15o; find,

(i) Each interior angle of the polygon.

(ii) Each exterior angle of the polygon.

(iii) Number of sides of the polygon.

Solution 8

(i)

Let each angle of measure x degree.

Therefore measure of each angle will be:

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

(ii)

Let each angle of measure x degree.

Therefore measure of each exterior angle will be:

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

(iii)

Let the number of each sides is n.

Now we can write

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Thus the number of sides are 12.

Question 9

The ratio between an exterior angle and an interior angle of a regular polygon is 2 : 3. Find the number of sides in the polygon.

Solution 9

Let measure of each interior and exterior angles are 3k and 2k.

Let number of sides of the polygon is n.

Now we can write:

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Again

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

From (1)

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Thus the number of sides of the polygon is 5.

Question 10

The difference between an exterior angle of (n - 1) sided regular polygon and an exterior angle of (n + 2) sided regular polygon is 6o find the value of n.

Solution 10

For (n-1) sided regular polygon:

Let measure of each angle is x.

Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

For (n+1) sided regular polygon:

Let measure of each angle is y.

Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Now we have

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Thus the value of n is 13.

Question 11

Two alternate sides of a regular polygon, when produced, meet at right angle. Find:

(i) The value of each exterior angle of the polygon;

(ii) The number of sides in the polygon.

Solution 11

(i)

Let the measure of each exterior angle is x and the number of sides is n.

Therefore we can write:

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Now we have

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

(ii)

Thus the number of sides in the polygon is:

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Chapter 14 - Rectilinear Figures [Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium] Exercise Ex. 14(B)

Question 1

State, 'true' or 'false'

(i) The diagonals of a rectangle bisect each other.

(ii) The diagonals of a quadrilateral bisect each other.

(iii) The diagonals of a parallelogram bisect each other at right angle.

(iv) Each diagonal of a rhombus bisects it.

(v) The quadrilateral, whose four sides are equal, is a square.

(vi) Every rhombus is a parallelogram.

(vii) Every parallelogram is a rhombus.

(viii) Diagonals of a rhombus are equal.

(ix) If two adjacent sides of a parallelogram are equal, it is a rhombus.

(x) If the diagonals of a quadrilateral bisect each other at right angle, the quadrilateral is a square.

Solution 1

(i)True.

This is true, because we know that a rectangle is a parallelogram. So, all the properties of a parallelogram are true for a rectangle. Since the diagonals of a parallelogram bisect each other, the same holds true for a rectangle.

(ii)False

This is not true for any random quadrilateral. Observe the quadrilateral shown below.

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Clearly the diagonals of the given quadrilateral do not bisect each other. However, if the quadrilateral was a special quadrilateral like a parallelogram, this would hold true.

(iii)False

Consider a rectangle as shown below.

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

It is a parallelogram. However, the diagonals of a rectangle do not intersect at right angles, even though they bisect each other.


(iv)True

Since a rhombus is a parallelogram, and we know that the diagonals of a parallelogram bisect each other, hence the diagonals of a rhombus too, bisect other.

(v)False

This need not be true, since if the angles of the quadrilateral are not right angles, the quadrilateral would be a rhombus rather than a square.

(vi)True

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

A parallelogram is a quadrilateral with opposite sides parallel and equal.

Since opposite sides of a rhombus are parallel, and all the sides of the rhombus are equal, a rhombus is a parallelogram.

(vii)False

This is false, since a parallelogram in general does not have all its sides equal. Only opposite sides of a parallelogram are equal. However, a rhombus has all its sides equal. So, every parallelogram cannot be a rhombus, except those parallelograms that have all equal sides.

(viii)False

This is a property of a rhombus. The diagonals of a rhombus need not be equal.

(ix)True

A parallelogram is a quadrilateral with opposite sides parallel and equal.

A rhombus is a quadrilateral with opposite sides parallel, and all sides equal.

If in a parallelogram the adjacent sides are equal, it means all the sides of the parallelogram are equal, thus forming a rhombus.

(x)False

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Observe the above figure. The diagonals of the quadrilateral shown above bisect each other at right angles, however the quadrilateral need not be a square, since the angles of the quadrilateral are clearly not right angles.

Question 2

In the figure, given below, AM bisects angle A and DM bisects angle D of parallelogram ABCD. Prove that : Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumAMD = 90o.

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Solution 2

From the given figure we conclude that

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Again from the Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Hence Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Question 3

In the following figure, AE and BC are equal and parallel and the three sides AB, CD and DE are equal to one another. If angle A is 102o. Find angles AEC and BCD.

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Solution 3

In the given figure

 

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

 

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Question 4

In a square ABCD, diagonals meet at O. P is a point on BC such that OB = BP.

Show that:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumPOC = Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumBDC = 2 Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumPOC

(iii) Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumBOP = 3 Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumCPO

Solution 4

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Question 5

The given figure shows a square ABCD and an equilateral triangle ABP. Calculate:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumAOB

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumBPC

(iii) Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumPCD

(iv) Reflex Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumAPC

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Solution 5

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

 

In the given figure Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumis an equilateral triangle

Therefore all its angles are Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Again in the

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Again

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Now

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium


(i)Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

(ii)Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

(iii)Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

(iv)Reflex Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Question 6

In the given figure ABCD is a rhombus with angle A = 67o

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

If DEC is an equilateral triangle, calculate:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumCBE(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumDBE.

Solution 6

Given that the figure ABCD is a rhombus with angle A = 67o

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium



Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

 

Question 7

In each of the following figures, ABCD is a parallelogram.

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium


In each case, given above, find the values of x and y.

Solution 7

(i)ABCD is a parallelogram

Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Thus

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Solving equations (i) and (ii) we have

x=5

y=3


(ii)

In the figure ABCD is a parallelogram

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Solving (i), (ii) we have

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Question 8

The angles of a quadrilateral are in the ratio 3 : 4 : 5 : 6. Show that the quadrilateral is a trapezium.

Solution 8

Given that the angles of a quadrilateral are in the ratio Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumLet the angles be Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Therefore the angles are

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Since all the angles are of different degrees thus forms a trapezium

Question 9

In a parallelogram ABCD, AB = 20 cm and AD = 12 cm. The bisector of angle A meets DC at E and BC produced at F. Find the length of CF.

 

Solution 9

 

 

 

 

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium 

 

 

 

 

 

 

 

 

Given AB = 20 cm and AD = 12 cm.

 

 

From the above figure, it's evident that ABF is an isosceles triangle with angle BAF = angle BFA = x

 

 

So AB = BF = 20

 

 

BF = 20

 

 

BC + CF = 20

 

 

CF = 20 - 12 = 8 cm

 

Question 10

In parallelogram ABCD, AP and AQ are perpendiculars from vertex of obtuse angle A as shown. If Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium find angles of the parallelogram.

 

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

 

 

 

 

 

Solution 10

We know that AQCP is a quadrilateral. So sum of all angles must be 360.

x + y + 90 + 90 = 360

x + y = 180

Given x:y = 2:1

So substitute x = 2y

3y = 180

y = 60

x = 120

We know that angle C = angle A = x = 120


Angle D = Angle B = 180 - x = 180 - 120 = 60

 

Hence, angles of parallelogram are 120, 60, 120 and 60. 

 

 

 

 

Chapter 14 - Rectilinear Figures [Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium] Exercise Ex. 14(C)

Question 1

E is the mid-point of side AB and F is the mid point of side DC of parallelogram ABCD. Prove that AEFD is a parallelogram.

Solution 1

Let us draw a parallelogram Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumWhere F is the midpoint

Of side DC of parallelogram Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

To prove:Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium is a parallelogram

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Proof: 

Therefore Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Also AD|| EF

therefore AEFC is a parallelogram.

Question 2

The diagonal BD of a parallelogram ABCD bisects angles B and D. Prove that ABCD is a rhombus.

Solution 2

GIVEN:Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium is a parallelogram where the diagonal Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium bisects

parallelogram Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium at angle Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

TO PROVE: Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium is a rhombus

Proof : Let us draw a parallelogram Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium where the diagonal Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium bisects the parallelogram at angle Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium.

Consruction :Let us join AC as a diagonal of the parallelogram

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Since Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium is a parallelogram

Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Diagonal Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium bisects angle Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

So Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Again Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium also bisects at Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Therefore Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Thus Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium is a rhombus.

Hence proved

Question 3

The alongside figure shows a parallelogram ABCD in which AE = EF = FC. Prove that:

(i) DE is parallel to FB

(ii) DE = FB

(iii) DEBF is a parallelogram.

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Solution 3

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Question 4

In the alongside diagram, ABCD is a parallelogram in which AP bisects angle AP bisects angle A and BQ bisects angle B. Prove that :

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

(i) AQ = BP

(ii) PQ = CD.

(iii) ABPQ is a parallelogram.

Solution 4

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium


Let us join PQ. 


Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

 

Question 5

In the given figure, ABCD is a parallelogram. Prove that: AB = 2 BC.

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Solution 5

Given Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumis a parallelogram

To prove:Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

 

Proof: Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumis a parallelogram

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Again,

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

NOW

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Hence proved

Question 6

Prove that the bisectors of opposite angles of a parallelogram are parallel.

Solution 6

Given ABCD is a parallelogram. The bisectors of Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumandSelina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium meet at EThe bisectors of Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumandSelina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium meet at F

From the parallelogram Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumwe have

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

In triangle ECD sum of anglesSelina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Similarly taking triangle Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumit can be prove that Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium


Now since

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Therefore the lines Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumand BF are parallel

Hence proved

Question 7

Prove that the bisectors of interior angles of a parallelogram form a rectangle.

Solution 7

Given:Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium is a parallelogram

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

TO PROVE:Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium is a rectangle

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Proof :

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumis a right triangle because its acute interior angles are complementary.

Similarly

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

since 3 angles of quadrilateral Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumare right angles,si is the 4th one and so Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumis a rectangle ,since its interior angles are all right angles

Hence proved

Question 8

Prove that the bisectors of the interior angles of a rectangle form a square.

Solution 8

Given: A parallelogram Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumin which Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Are the bisects of Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumrespectively forming quadrilateralsSelina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium.

To prove: Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumis a rectangle

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Proof :

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumAlso in

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

From the above equation we get

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Hence PQRS is a rectangle

Question 9

In parallelogram ABCD, the bisector of angle A meets DC at P and AB = 2 AD.

Prove that:

(i) BP bisects angle B.(ii) Angle APB = 90o.

Solution 9

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium


(i)LetSelina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Also Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumis the bisector Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

 

Now ,

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumSelina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Therefore Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Now

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Also ,Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

In Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Therefore Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Hence Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumbisect Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

(ii)

Opposite angles are supplementary

Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Hence proved

Question 10

Points M and N are taken on the diagonal AC of a parallelogram ABCD such that AM = CN. Prove that BMDN is a parallelogram.

Solution 10

Points Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumare taken on the diagonal Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumof a parallelogram Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumsuch that Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium.

Prove that Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumis a parallelogram

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

CONSTRUCTION: Join Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumto meet Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumin Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium.

PROOF: We know that the diagonals of parallelogram bisect each other.

Now,Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumbisect each other at Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium.

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Thus in a quadrilateral Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium,diagonal Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumare such that Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumand Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Therefore the diagonals Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumbisect each other.

Hence Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumis a parallelogram

Question 11

In the following figure, ABCD is a parallelogram. Prove that:

(i) AP bisects angle A

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

(ii) BP bisects angle B

(iii) Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumDAP + Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumBCP = Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumAPB

Solution 11

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Consider  Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Therefore Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Hence proved

Question 12

ABCD is a square. A is joined to a point P on BC and D is joined to a point Q on AB. If AP = DQ; prove that AP and DQ are perpendicular to each other.

Solution 12

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

ABCD is a square and Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Question 13

In a quadrilateral ABCD, AB = AD and CB = CD. Prove that :

(i) AC bisects angle BAD.

(ii) AC is perpendicular bisector of BD.

Solution 13

Given: Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumis quadrilateral,

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

To prove: (i) AC bisects angle BAD.

(ii) AC is perpendicular bisector of BD.

Proof :

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Therefore Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumbisects Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumSelina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Thus Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumis perpendicular bisector of Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Hence proved

Question 14

The following figure shows a trapezium ABCD in which AB is parallel to DC and AD = BC. Prove that:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumDAB = Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumCBA

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumADC = Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumBCD

(iii) AC = BD

(iv) OA = OB and OC = OD

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Solution 14

Given Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium is a trapezium,Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

To prove(i)Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumDAB = Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumCBA

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumADC = Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumBCD

(iii) AC = BD

(iv) OA = OB and OC = OD

Proof :(i) Since Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium and transversal Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumcuts them at Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumrespectively.

Therefore, Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Since Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Therefore Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium is a parallelogram

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

In Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium,we have

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Since Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Again Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Hence proved

Question 15

In the given figure, AP is bisector of Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumA and CQ is bisector of Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumC of parallelogram ABCD. Prove that APCQ is a parallelogram.

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Solution 15

 Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Question 16

In case of a parallelogram prove that:

(i) The bisectors of any two adjacent angles intersect at 90o.

(ii) The bisectors of opposite angles are parallel to each other.

Solution 16

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumis a parallelogram ,the bisectors of Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumand

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziummeet at a point Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumand the bisectors of Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And TrapeziumAND Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziummeet at F.

We have to prove that the Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapeziumand Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Proof :In the parallelogram Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

 

Similarly taking triangle BCF it can be proved thatSelina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Also Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Now since Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Hence proved

Question 17

The diagonals of a rectangle intersect each other at right angles. Prove that the rectangle is a square.

Solution 17

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium 

Question 18

In the following figure, ABCD and PQRS are two parallelograms such that D = 120° and Q = 70°. Find the value of x.

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium 

Solution 18

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium

Question 19

In the following figure, ABCD is a rhombus and DCFE is a square.

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium 

If ABC =56°, find:

(i) DAE

(ii) FEA

(iii) EAC

(iv) AEC

Solution 19

Selina Solutions Icse Class 9 Mathematics Chapter - Rectilinear Figures Quadrilaterals Parallelogram Rectangle Rhombus Square And Trapezium 

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