ICSE Class 9 Rectilinear Figures Solution New Pattern

## ICSE Class 9 Rectilinear Figures Solution New Pattern By Clarify Knowledge

ICSE Class 9 Rectilinear Figures Solution New Pattern 2022

## 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

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

Hence the angles of the pentagon are:

4×20= 80, 8×20= 160, 6×20= 120, 4×20= 80, 5×20= 100Question 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

Therefore the measure of each equal angles are 116Question 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:

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:

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:

Thus the measure of each equal angles are 127o.Question 7

In a pentagon ABCDE, AB is parallel to DC and A : E : D = 3 : 4 : 5. Find angle E.Solution 7

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

Thus

Thus the measure of angle E will be 4×30=120Question 8

AB, BC and CD are the three consecutive sides of a regular polygon. If BAC = 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:

(ii)

Let each angle of measure x degree.

Therefore measure of each exterior angle will be:

(iii)

Let the number of each sides is n.

Now we can write

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:

Again

From (1)

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

For (n+1) sided regular polygon:

Let measure of each angle is y.

Therefore

Now we have

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:

Now we have

(ii)

Thus the number of sides in the polygon is:

## 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.

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.

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

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

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 : AMD = 90o.

Solution 2

From the given figure we conclude that

Again from the

Hence 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.

Solution 3

In the given figure

Question 4

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

Show that:

(i) POC =

(ii) BDC = 2 POC

(iii) BOP = 3 CPOSolution 4

Question 5

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

(i) AOB

(ii) BPC

(iii) PCD

(iv) Reflex APC

Solution 5

In the given figure is an equilateral triangle

Therefore all its angles are

Again in the

Again

Now

Therefore

(i)

(ii)

(iii)

(iv)Reflex Question 6

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

If DEC is an equilateral triangle, calculate:

(i) CBE(ii) DBE.Solution 6

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

Question 7

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

(i)

(ii)

In each case, given above, find the values of x and y.Solution 7

(i)ABCD is a parallelogram

Therefore

Thus

Solving equations (i) and (ii) we have

x=5

y=3

(ii)

In the figure ABCD is a parallelogram

Therefore

Solving (i), (ii) we have

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 Let the angles be

Therefore the angles are

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

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 cmQuestion 10

In parallelogram ABCD, AP and AQ are perpendiculars from vertex of obtuse angle A as shown. If  find angles of the parallelogram.

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 Where F is the midpoint

Of side DC of parallelogram

To prove: is a parallelogram

Proof:

Therefore

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: is a parallelogram where the diagonal  bisects

parallelogram  at angle

TO PROVE:  is a rhombus

Proof : Let us draw a parallelogram  where the diagonal  bisects the parallelogram at angle .

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

Since  is a parallelogram

Therefore

Diagonal  bisects angle

So

Again  also bisects at

Therefore

Thus  is a rhombus.

Hence provedQuestion 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.

Solution 3

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 :

(i) AQ = BP

(ii) PQ = CD.

(iii) ABPQ is a parallelogram.Solution 4

Let us join PQ.

Question 5

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

Solution 5

Given is a parallelogram

To prove:

Proof: is a parallelogram

Again,

NOW

Hence provedQuestion 6

Prove that the bisectors of opposite angles of a parallelogram are parallel.Solution 6

Given ABCD is a parallelogram. The bisectors of and meet at E. The bisectors of and meet at F

From the parallelogram we have

In triangle ECD sum of angles

Similarly taking triangle it can be prove that

Now since

Therefore the lines and BF are parallel

Hence provedQuestion 7

Prove that the bisectors of interior angles of a parallelogram form a rectangle.Solution 7

Given: is a parallelogram

TO PROVE: is a rectangle

Proof :

is a right triangle because its acute interior angles are complementary.

Similarly

since 3 angles of quadrilateral are right angles,si is the 4th one and so is a rectangle ,since its interior angles are all right angles

Hence provedQuestion 8

Prove that the bisectors of the interior angles of a rectangle form a square.Solution 8

Given: A parallelogram in which

Are the bisects of respectively forming quadrilaterals.

To prove: is a rectangle

Proof :

Also in

From the above equation we get

Hence PQRS is a rectangleQuestion 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

(i)Let

Also is the bisector

Now ,

Therefore

Now

Therefore

Also ,

In

Therefore

Hence bisect

(ii)

Opposite angles are supplementary

Therefore

Hence provedQuestion 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 are taken on the diagonal of a parallelogram such that .

Prove that is a parallelogram

CONSTRUCTION: Join to meet in .

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

Now,bisect each other at .

Thus in a quadrilateral ,diagonal are such that and

Therefore the diagonals bisect each other.

Hence is a parallelogramQuestion 11

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

(i) AP bisects angle A

(ii) BP bisects angle B

(iii) DAP + BCP = APBSolution 11

Consider

Therefore

Hence provedQuestion 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

ABCD is a square and

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: is quadrilateral,

To prove: (i) AC bisects angle BAD.

(ii) AC is perpendicular bisector of BD.

Proof :

Therefore bisects

Thus is perpendicular bisector of

Hence provedQuestion 14

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

(i) DAB = CBA

(ii) ADC = BCD

(iii) AC = BD

(iv) OA = OB and OC = OD

Solution 14

Given  is a trapezium,

To prove(i)DAB = CBA

(ii) ADC = BCD

(iii) AC = BD

(iv) OA = OB and OC = OD

Proof :(i) Since  and transversal cuts them at respectively.

Therefore,

Since

Therefore  is a parallelogram

In ,we have

Since

Again

Hence provedQuestion 15

In the given figure, AP is bisector of A and CQ is bisector of C of parallelogram ABCD. Prove that APCQ is a parallelogram.

Solution 15

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

is a parallelogram ,the bisectors of and

meet at a point and the bisectors of AND meet at F.

We have to prove that the and

Proof :In the parallelogram

Similarly taking triangle BCF it can be proved that

Also

Now since

Hence provedQuestion 17

The diagonals of a rectangle intersect each other at right angles. Prove that the rectangle is a square.Solution 17

Question 18

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

Solution 18

Question 19

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

If ∠ABC =56°, find:

(i) ∠DAE

(ii) ∠FEA

(iii) ∠EAC

(iv) ∠AECSolution 19

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