ICSE

ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 13 Understanding Quadrilaterals Check Your Progress

Question 1.
From the given diagram, find the value of x.

Solution:

Question 2.
If two angles of a quadrilateral are 77° and 51°, and out of the remaining two angles, one angle is 10° smaller than the other, find these angles.
Solution:

Question 3.
In the given figure, AB || DC, ∠A = 74° and ∠B : ∠C = 4 : 5. Find
(i) ∠D
(ii) ∠B
(iii) ∠C

Solution:

Question 4.
In quadrilateral ABCD, ∠A : ∠B : ∠C : ∠D = 3 : 4 : 6 : 7. Find all the angles of the quadrilateral. Hence, prove that AB and DC are parallel. Is BC also parallel to AD?
Solution:

Question 5.
One angle of a parallelogram is two-third of the other. Find the angles of the parallelogram.
Solution:

Question 6.
In the given figure, ABCD is a kite. If ∠BCD = 52° and ∠ADB = 42°, find the values of x, y, and z.

Solution:

Question 7.
In the given figure, ABCD is a rectangle. Prove that AC = BD.

Solution:

Question 8.
In the given figure, ABCD is a rhombus and EDC is an equilateral triangle. If ∠DAB = 48°, find
(i) ∠BEC
(ii) ∠DEB
(iii) ∠BFC

Solution:

Question 9.
Find the number of sides of a regular polygon if each of its interior angles is 168°.
Solution:

Question 10.
If the sum of interior angles of a polygon is 3780° find the number of sides.
Solution:

Question 11.
The angles of a hexagon are (2x + 5)°, (3x – 5)°, (x + 40)°, (2x + 20)°, (2x + 25)° and (2x + 35)°. Find the value of x.
Solution:

Question 12.
Two angles of a polygon are right angles and every other angle is 120°. Find the number of sides of the polygon.
Solution:

Question 13.
The sum of interior angles of a regular polygon is twice the sum of its exterior angles. Find the number of sides of the polygon.
Solution:

Question 14.
An exterior angle of a regular polygon is one- fourth of its interior angle. Find the number of sides in the polygon.
Solution:

ML Aggarwal Class 8 Solutions for ICSE Maths

ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 13 Understanding Quadrilaterals Objective Type Questions

Mental Maths
Question 1.
Fill in the blanks:
(i) The curves which have different beginning and endpoints are called …………
(ii) A curve which does not cross itself at any point is called a …………
(iii) A simple closed curve made up entirely of line segments is called a …………
(iv) A polygon in which each interior angle is less than 180° is called a …………
(v) 10 sided polygon is called …..
(vi) A polygon which has all its sides of equal length and all its angles of equal measure is called a …………
(vii) Sum of all exterior angles of a polygon is …………
(vi) A polygon which has all its sides of equal length and all its angles of equal measure is called a …………
(vii) Sum of all exterior angles of a polygon is …………
(viii) Sum of all interior angles of a n-sided polygon is …………
(ix) The adjacent angles of a parallelogram are …………
(x) If an angle of a parallelogram is a right angle, then it is called a …………
(xi) If two adjacent sides of a parallelogram are equal, then it is called a …………
(xii) It two adjacent sides of a rectangle are equal, then it is called a …………
(xiii) The diagonal of a rhombus bisect each other at …………
(xiv) A quadrilateral in which one pair of opposite sides is parallel is called a …………
(xv) A quadrilateral in which two pairs of adjacent sides are equal is called a …………
(xvi) If two non-parallel sides of a trapezium are equal then it is called …………
Solution:

Question 2.
State whether the following statements are true (T) or false (F):
(i) The curves which have the same beginning and endpoints are called open curves.
(ii) The region of the plane that lies inside the curve is called the interior of the curve.
(iii) A polygon in which at least one interior angle is greater than 180° is called convex polygon.
(iv) 6 sided polygon is called hexagon.
(v) Sum of all interior angles of a quadrilateral is 180°.
(vi) Each interior angle of a n-sided regular polygon is \(\frac{(2 n-4) \times 90^{\circ}}{n}\).
(vii) The diagonals of a parallelogram bisect each other at right angles.
(viii) The opposite angles of a parallelogram are of equal measure.
(ix) The diagonals of a rhombus bisect the angles of a rhombus.
(x) The diagonals of a square are not equal.
(xi) Co-interior angles of a parallelogram are supplementary.
(xii) The diagonals of a kite bisect at right angles.
(xiii) All rectangles are squares.
(xiv) All rhombuses are parallelograms.
(xv) All squares are rhombuses and also rectangles.
(xvi) All squares are not parallelograms.
(xvii) All kites are rhombuses.
(xviii) All rhombuses are kites.
(xix) All parallelograms are trapeziums.
(xx) All squares are trapeziums.
Solution:

Multiple Choice Questions
Choose the correct answer from the given four options (3 to 19):
Question 3.
Sum of all interior angles of an 11-sided polygon is
(a) 1620°
(b) 1440°
(c) 1260v
(d) none of these
Solution:

Question 4.
If each interior angle of a regular polygon is 144°, then number of sides of polygon is
(a) 8
(b) 9
(c) 10
(d) 11
Solution:

Question 5.
If the sum of all interior angles of a polygon is 1260°, then number of sides of polygon is
(a) 6
(b) 7
(c) 8
(d) 9
Solution:

Question 6.
The sum of all exterior angles of a pentagon is
(a) 590°
(b) 360°
(c) 180°
(d) none of these
Solution:

Question 7.
If the ratio between an exterior and interior angle of a regular polygon is 1 : 5, then the number of sides of the polygon is
(a) 11
(b) 12
(c) 13
(d) 14
Solution:

Question 8.
In the given figure, the value of x is
(a) 140°
(b) 50°
(c) 130°
(d) 40°

Solution:

Question 9.
In the given figure, the value of x is
(a) 120°
(b) 130°
(c) 140°
(d) 150°

Solution:

Question 10.
In the given figure, the value of x + y + z + w is
(a) 180°
(b) 270°
(c) 300°
(d) 360°

Solution:

Question 11.
In the given figure, the value of x + y is
(a) 180°
(b) 190°
(c) 170°
(d) 160°

Solution:

Question 12.
The lengths of two adjacent sides of a parallelogram are in the ratio 1 : 2. If the perimeter of a parallelogram is 60 cm, then the length of its sides are
(a) 6 cm, 12 cm
(b) 8 cm, 16 cm
(c) 9 cm, 18 cm
(d) 10 cm, 20 cm
Solution:

Question 13.
In the given figure, ABCD is a parallelogram, the values of x and y respectively are
(a) 1 cm, 1 cm
(b) 2 cm, 1 cm
(c) 1 cm, 2 cm
(d) 2 cm, 2 cm

Solution:

Question 14.
In the given figure, ABCD is a parallelogram, the values of x, y and z respectively are
(a) 60°, 60°, 70°
(b) 60°, 70°, 60°
(c) 70°, 60°, 60°
(d) none of these

Solution:

Question 15.
In the given figure, ABCD is a rectangle, the value of angle x is
(a) 60°
(b) 90°
(c) 120°
(d) none of these

Solution:

Question 16.
In a rhombus ABCD, the diagonals AC and BD are respectively 8 cm and 6 cm. The length of each side of the rhombus is
(a) 7 cm
(b) 5 cm
(c) 6 cm
(d) 8 cm
Solution:

Question 17.
In the given figure, ABCD is a square, the value of angle x is
(a) 30°
(b) 45°
(c) 60°
(d) not possible to find

Solution:

Question 18.
In the given figure, ABCD is a kite, the value of angle x is
(a) 86°
(b) 100°
(c) 104°
(d) none of these

Solution:

Question 19.
In the given figure, ABCD is an isosceles trapezium. The values of x, y and z respectively are
(a) 110°, 110°, 70°
(b) 110°, 70°, 110°
(c) 70°, 110°, 110°
(d) none of these

Solution:

Higher Order Thinking Skills (Hots)
Question 1.
In the given figure, ABCDEF is a regular hexagon. Prove that quadrilaterals ABDE and ACDF are parallelograms. Also prove that quadrilateral AGDH is a parallelogram.

Solution:

Question 2.
In the given figure, ABCD is a parallelogram and M, N are the mid-points of sides BC, AD respectively. Prove that EA = AB = BF.

Solution:

Question 3.
Prove that the quadrilateral formed by joining the mid-points of the adjacent sides of a rectangle is a parallelogram.
Solution:

ML Aggarwal Class 8 Solutions for ICSE Maths

ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 13 Understanding Quadrilaterals Ex 13.3

Question 1.
Identify all the quadrilaterals that have
(i) four sides of equal length
(ii) four right angles.
Solution:

Question 2.
Explain how a square is
(i) a quadrilateral
(ii) a parallelogram
(iii) a rhombus
(iv) a rectangle.
Solution:

Question 3.
Name the quadrilaterals whose diagonals
(i) bisect each other
(ii) are perpendicular bisectors of each other
(iii) are equal.
Solution:

Question 4.
One of the diagonals of a rhombus and its sides are equal. Find the angles of the rhombus.
Solution:

Question 5.
In the given figure, ABCD is a rhombus, find the values of x, y and z.

Solution:

Question 6.
In the given figure, ABCD is a trapezium. If ∠A : ∠D = 5 : 7, ∠B = (3x + 11)° and ZC = (5x – 31)°, then find all the angles of the trapezium.

Solution:

Question 7.
In the given figure, ABCD is a rectangle. If ∠CEB : ∠ECB = 3 : 2 find
(i) ∠CEB,
(ii) ∠DCF

Solution:

Question 8.
In the given figure, ABCD is a rectangle and diagonals intersect at O. If ∠AOB = 118°, find

(i) ∠ABO
(ii) ∠ADO
(iii) ∠OCB
Solution:

Question 9.
In the given figure, ABCD is a rhombus and ∠ABD = 50°. Find :
(i) ∠CAB
(ii) ∠BCD
(iii) ∠ADC

Solution:

Question 10.
In the given isosceles trapezium ABCD, ∠C = 102°. Find all the remaining angles of the trapezium.

Solution:

Question 11.
In the given figure, PQRS is a kite. Find the values of x and y.

Solution:

ML Aggarwal Class 8 Solutions for ICSE Maths

ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 13 Understanding Quadrilaterals Ex 13.2

Question 1.
In the given figure, ABCD is a parallelogram. Complete each statement along with the definition or property used.
(i) AD = ………..
(ii) DC = ………..
(iii) ∠DCB = ………..
(iv) ∠ADC = ………..
(v) ∠DAB = ………..
(vi) OC = ………..
(vii) OB = ………..
(viii) m∠DAB + m∠CDA = ………..

Solution:

Question 2.
Consider the following parallelograms. Find the values of x, y, z in each.

Solution:

Question 3.
Two adjacent sides of a parallelogram are in the ratio 5 : 7. If the perimeter of a parallelogram is 72 cm, find the length of its sides.
Solution:

Question 4.
The measure of two adjacent angles of a parallelogram is in the ratio 4 : 5. Find the measure of each angle of the parallelogram.
Solution:

Question 5.
Can a quadrilateral ABCD be a parallelogram, give reasons in support of your answer.
(i) ∠A + ∠C= 180°?
(ii) AD = BC = 6 cm, AB = 5 cm, DC = 4.5 cm?
(iii) ∠B = 80°, ∠D = 70°?
(iv) ∠B + ∠C= 180°?
Solution:

Question 6.
In the following figures, HOPE and ROPE are parallelograms. Find the measures of angles x, y and z. State the properties you use to find them.

Solution:

Question 7.
In the given figure TURN and BURN are parallelograms. Find the measures of x and y (lengths are in cm).

Solution:

Question 8.
In the following figure, both ABCD and PQRS are parallelograms. Find the value of x.

Solution:

Question 9.
In the given figure, ABCD, is a parallelogram and diagonals intersect at O. Find :
(i) ∠CAD
(ii) ∠ACD
(iii) ∠ADC

Solution:

Question 10.
In the given figure, ABCD is a parallelogram. Perpendiculars DN and BP are drawn on diagonal AC. Prove that:
(i) ∆DCN ≅ ∆BAP
(ii) AN = CP

Solution:

Question 11.
In the given figure, ABC is a triangle. Through A, B and C lines are drawn parallel to BC, CA and AB respectively, which forms a ∆PQR. Show that 2(AB + BC + CA) = PQ + QR + RP.

Solution:

ML Aggarwal Class 8 Solutions for ICSE Maths

ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 13 Understanding Quadrilaterals Ex 13.1

Question 1.
Some figures are given below.

Classify each of them on the basis of the following:
(a) Simple curve
(b) Simple closed curve
(c) Polygon
(d) Convex polygon
(e) Concave polygon
Solution:

Question 2.
How many diagonals does each of the following have?
(a) A convex quadrilateral
(b) A regular hexagon
Solution:
(a) A convex quadrilateral: It has two diagonals.
(b) A regular hexagon: It has 9 diagonals as shown.

Question 3.
Find the sum of measures of all interior angles of a polygon with the number of sides:
(i) 8
(ii) 12
Solution:

Question 4.
Find the number of sides of a regular polygon whose each exterior angles has a measure of
(i) 24°
(ii) 60°
(iii) 72°
Solution:

Question 5.
Find the number of sides of a regular polygon if each of its interior angles is
(i) 90°
(ii) 108°
(iii) 165°
Solution:

Question 6.
Find the number of sides in a polygon if the sum of its interior angles is:
(i) 1260°
(ii) 1980°
(iii) 3420°
Solution:

Question 7.
If the angles of a pentagon are in the ratio 7 : 8 : 11 : 13 : 15, find the angles.
Solution:

Question 8.
The angles of a pentagon are x°, (x – 10)°, (x + 20)°, (2x – 44)° and (2x – 70°) Calculate x.
Solution:

Question 9.
The exterior angles of a pentagon are in ratio 1 : 2 : 3 : 4 : 5. Find all the interior angles of the pentagon.
Solution:

Question 10.
In a quadrilateral ABCD, AB || DC. If ∠A : ∠D = 2:3 and ∠B : ∠C = ∠7 : 8, find the measure of each angle.
Solution:

Question 11.
From the adjoining figure, find
(i) x
(ii) ∠DAB
(iii) ∠ADB

Solution:

Question 12.
Find the angle measure x in the following figures:

Solution:

Question 13.
(i) In the given figure, find x + y + z.

(ii) In the given figure, find x + y + z + w.

Solution:

Question 14.
A heptagon has three equal angles each of 120° and four equal angles. Find the size of equal angles.
Solution:

Question 15.
The ratio between an exterior angle and the interior angle of a regular polygon is 1 : 5. Find
(i) the measure of each exterior angle
(ii) the measure of each interior angle
(iii) the number of sides in the polygon.
Solution:

Question 16.
Each interior angle of a regular polygon is double of its exterior angle. Find the number of sides in the polygon.
Solution:

ML Aggarwal Class 8 Solutions for ICSE Maths