Rectilinear Figures Class-9th Concise Selina ICSE Maths Solutions Chapter-14. We provide step by step Solutions of Exercise / lesson-14 Rectilinear Figures (Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium) for ICSE Class-9th Concise Selina Mathematics by R K Bansal.
Our Solutions contain all type Questions with Exe-14 A, Exe-14 B, and Exe-14 C, to develop skill and confidence. Visit official Website CISCE for detail information about ICSE Board Class-9th Mathematics .
Rectilinear Figures Class-9th Concise Selina ICSE Maths Solutions Chapter-14 (Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium)
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Exercise – 14 A, Rectilinear Figures Class-9th Concise (Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium)
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.
Answer
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.
Answer
Let the angles of the pentagon are 4x, 8x, 6x, 4x and 5x.
Thus we can write
Question 3
One angle of a six-sided polygon is 140o and the other angles are equal. Find the measure of each equal angle.
Answer
Let the measure of each equal angles are x.
Then we can write
Therefore the measure of each equal angles are 116 ‘
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.
Answer
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.
Answer
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.
Answer
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.
Answer
Let the measure of the angles are 3x, 4x and 5x.
Thus
Thus the measure of angle E will be 4×30=120
Question 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
Answer
(i)
Let each angle of measure x degree.
Therefore measure of each angle will be:
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.
Answer
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:
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.
Answer
For (n-1) sided regular polygon:
Let measure of each angle is x.
Therefore
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.
Answer
(i)
Let the measure of each exterior angle is x and the number of sides is n.
Therefore we can write:
require number of side of regular polygon = 8
Concise Selina Solutions for ICSE Class-9 Maths Exercise – 14 B Rectilinear Figures (Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium)
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.
Answer
(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.
…………………
Answer
From the given figure we conclude that
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.
……………….
Answer
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 CPO
Answer
Question 5
The given figure shows a square ABCD and an equilateral triangle ABP. Calculate:
(i) ∠AOB
(ii) ∠BPC
(iii) ∠PCD
(iv) Reflex ∠APC
Answer
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.
Answer
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.
Answer
(i)ABCD is a parallelogram
Therefore
Question 8
The angles of a quadrilateral are in the ratio 3 : 4 : 5 : 6. Show that the quadrilateral is a trapezium.
Answer
Given that the angles of a quadrilateral are in the ratio Let the angles be
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.
Answer
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 find angles of the parallelogram.
Answer
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.
Concise Selina Solutions for ICSE Class-9 Maths Rectilinear Figures (Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium) Exercise – 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.
Answer
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.
Answer
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.
……
Answer
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.
Answer
Question 5
In the given figure, ABCD is a parallelogram. Prove that: AB = 2 BC.
Answer
Hence proved
Question 6
Prove that the bisectors of opposite angles of a parallelogram are parallel.
Answer
Question 7
Prove that the bisectors of interior angles of a parallelogram form a rectangle.
Answer
Question 8
Prove that the bisectors of the interior angles of a rectangle form a square.
Answer
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.
Answer
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.
Answer
Question 11
In the following figure, ABCD is a parallelogram. Prove that:
(i) AP bisects angle A
(ii) BP bisects angle B
(iii)……………….
Answer
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.
Answer
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.
Answer
Question 14
The following figure shows a trapezium ABCD in which AB is parallel to DC and AD = BC. Prove that:
(i) ………………
(ii) ………………
(iii) AC = BD
(iv) OA = OB and OC = OD
Answer
Question 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.
Answer
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.
Answer
Question 17
The diagonals of a rectangle intersect each other at right angles. Prove that the rectangle is a square.
Answer
Question 18
In the following figure, ABCD and PQRS are two parallelograms such that ∠D = 120° and ∠Q = 70°. Find the value of x.
Answer
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) ∠AEC
Answer
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