NCERT Solutions for Class 8 Maths Chapter 3: Understanding Quadrilaterals
NCERT Solutions for Class 8 Mathematics Chapter 3, "Understanding Quadrilaterals," are available here for free download in PDF format. These meticulously crafted solutions, developed by expert mathematicians at Orchid The International School, are designed with a focus on examination preparation.
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Students can access the NCERT Solutions for Class 8 Maths Chapter 3: Understanding Quadrilaterals. Curated by experts according to the CBSE syllabus for 2023–2024, these step-by-step solutions make Maths much easier to understand and learn for the students. These solutions can be used in practice by students to attain skills in solving problems, reinforce important learning objectives, and be well-prepared for tests.
Exercise 3.1
Given here are some figures.
Classify each of them on the basis of the following.
Simple curve (b) Simple closed curve (c) Polygon
(d) Convex polygon (e) Concave polygon
a) Simple curve: 1, 2, 5, 6 and 7
b) Simple closed curve: 1, 2, 5, 6 and 7
c) Polygon: 1 and 2
d) Convex polygon: 2
e) Concave polygon: 1
How many diagonals does each of the following have?
(a) A convex quadrilateral
(b) A regular hexagon
(c) A triangle
a) A convex quadrilateral: 2.
b) A regular hexagon: 9.
c) A triangle: 0
What is the sum of the measures of the angles of a convex quadrilateral? Will this property hold if the quadrilateral is not convex? (Make a non-convex quadrilateral and try!)
Let ABCD be a convex quadrilateral.
From the figure, we infer that the quadrilateral ABCD is formed by two triangles,
i.e. ΔADC and ΔABC.
Since we know that sum of the interior angles of a triangle is 180°,
the sum of the measures of the angles is 180° + 180° = 360°
Let us take another quadrilateral ABCD which is not convex .
Join BC, such that it divides ABCD into two triangles ΔABC and ΔBCD. In ΔABC,
∠1 + ∠2 + ∠3 = 180° (angle sum property of triangle)
In ΔBCD,
∠4 + ∠5 + ∠6 = 180° (angle sum property of triangle)
∴, ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 180° + 180°
⇒ ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 360°
⇒ ∠A + ∠B + ∠C + ∠D = 360°
Thus, this property holds if the quadrilateral is not convex.
Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that.)
What can you say about the angle sum of a convex polygon with number of sides? (a) 7 (b) 8 (c) 10 (d) n
The angle sum of a polygon having side n = (n-2)×180°
a) 7
Here, n = 7
Thus, angle sum = (7-2)×180° = 5×180° = 900°
b) 8
Here, n = 8
Thus, angle sum = (8-2)×180° = 6×180° = 1080°
c) 10
Here, n = 10
Thus, angle sum = (10-2)×180° = 8×180° = 1440°
d) n
Here, n = n
Thus, angle sum = (n-2)×180°
What is a regular polygon?
State the name of a regular polygon of
(i) 3 sides
(ii) 4 sides
(iii) 6 sides
Regular polygon: A polygon having sides of equal length and angles of equal measures is called a regular polygon. A regular polygon is both equilateral and equiangular.
(i) A regular polygon of 3 sides is called an equilateral triangle.
(ii) A regular polygon of 4 sides is called a square.
(iii) A regular polygon of 6 sides is called a regular hexagon.
Find the angle measure of x in the following figures.
a) The figure has 4 sides. Hence, it is a quadrilateral. Sum of angles of the quadrilateral = 360°
⇒ 50° + 130° + 120° + x = 360°
⇒ 300° + x = 360°
⇒ x = 360° – 300° = 60°
b) The figure has 4 sides. Hence, it is a quadrilateral. Also, one side is perpendicular forming a right angle.
Sum of angles of the quadrilateral = 360°
⇒ 90° + 70° + 60° + x = 360°
⇒ 220° + x = 360°
⇒ x = 360° – 220° = 140°
c) The figure has 5 sides. Hence, it is a pentagon.
Sum of angles of the pentagon = 540° Two angles at the bottom are a linear pair.
∴, 180° – 70° = 110°
180° – 60° = 120°
⇒ 30° + 110° + 120° + x + x = 540°
⇒ 260° + 2x = 540°
⇒ 2x = 540° – 260° = 280°
⇒ 2x = 280°
= 140°
d) The figure has 5 equal sides. Hence, it is a regular pentagon. Thus, all its angles are equal.
5x = 540°
⇒ x = 540°/5
⇒ x = 108°
a) Sum of all angles of triangle = 180°
One side of triangle = 180°- (90° + 30°) = 60°
x + 90° = 180° ⇒ x = 180° – 90° = 90°
y + 60° = 180° ⇒ y = 180° – 60° = 120°
z + 30° = 180° ⇒ z = 180° – 30° = 150°
x + y + z = 90° + 120° + 150° = 360°
b) Sum of all angles of quadrilateral = 360°
One side of quadrilateral = 360°- (60° + 80° + 120°) = 360° – 260° = 100°
x + 120° = 180° ⇒ x = 180° – 120° = 60°
y + 80° = 180° ⇒ y = 180° – 80° = 100°
z + 60° = 180° ⇒ z = 180° – 60° = 120°
w + 100° = 180° ⇒ w = 180° – 100° = 80°
x + y + z + w = 60° + 100° + 120° + 80° = 360°
Exercise 3.2
a) What is the minimum interior angle possible for a regular polygon? Why?
b) What is the maximum exterior angle possible for a regular polygon?
a) An equilateral triangle is the regular polygon (with 3 sides) having the least possible minimum interior angle because a regular polygon can be constructed with minimum 3 sides.
Since the sum of interior angles of a triangle = 180°
Each interior angle = 180/3 = 60°
b) An equilateral triangle is the regular polygon (with 3 sides) having the maximum exterior angle because the regular polygon with the least number of sides has the maximum exterior angle possible. Maximum exterior possible = 180 – 60° = 120°
Find x in the following figures.
a)
125° + m = 180° ⇒ m = 180° – 125° = 55° (Linear pair)
125° + n = 180° ⇒ n = 180° – 125° = 55° (Linear pair)
x = m + n (The exterior angle of a triangle is equal to the sum of the two opposite interior angles)
⇒ x = 55° + 55° = 110°
b)
Two interior angles are right angles = 90°
70° + m = 180° ⇒ m = 180° – 70° = 110° (Linear pair)
60° + n = 180° ⇒ n = 180° – 60° = 120° (Linear pair) The figure is having five sides and is a pentagon.
Thus, sum of the angles of a pentagon = 540°
⇒ 90° + 90° + 110° + 120° + y = 540°
⇒ 410° + y = 540° ⇒ y = 540° – 410° = 130°
x + y = 180° (Linear pair)
⇒ x + 130° = 180°
⇒ x = 180° – 130° = 50°
Find the measure of each exterior angle of a regular polygon of
(i) 9 sides
(ii) 15 sides
Sum of the angles of a regular polygon having side n = (n-2)×180°
(i) Sum of the angles of a regular polygon having 9 sides = (9-2)×180°= 7×180° = 1260°
Each interior angle=1260/9 = 140°
Each exterior angle = 180° – 140° = 40°
Or,
Each exterior angle = Sum of exterior angles/Number of angles = 360/9 = 40°
(ii) Sum of angles of a regular polygon having side 15 = (15-2)×180°
= 13×180° = 2340°
Each interior angle = 2340/15 = 156°
Each exterior angle = 180° – 156° = 24°
Or,
Each exterior angle = sum of exterior angles/Number of angles = 360/15 = 24°
How many sides does a regular polygon have if the measure of an exterior angle is 24°?
Each exterior angle = sum of exterior angles/Number of angles
24°= 360/ Number of sides
⇒ Number of sides = 360/24 = 15
Thus, the regular polygon has 15 sides.
How many sides does a regular polygon have if each of its interior angles is 165°?
Interior angle = 165°
Exterior angle = 180° – 165° = 15°
Number of sides = sum of exterior angles/exterior angles
⇒ Number of sides = 360/15 = 24
Thus, the regular polygon has 24 sides.
a) Is it possible to have a regular polygon with measure of each exterior angle as 22°?
b) Can it be an interior angle of a regular polygon? Why?
a) Exterior angle = 22°
Number of sides = sum of exterior angles/ exterior angle
⇒ Number of sides = 360/22 = 16.36
No, we can’t have a regular polygon with each exterior angle as 22° as it is not a divisor of 360.
b) Interior angle = 22°
Exterior angle = 180° – 22°= 158°
No, we can’t have a regular polygon with each exterior angle as 158° as it is not a divisor of 360.
Exercise 3.3
Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.
ABCD is a figure of quadrilateral that is not a parallelogram but has exactly two opposite angles, that is, ∠B = ∠D of equal measure. It is not a parallelogram because ∠A ≠ ∠C.
The measures of two adjacent angles of a parallelogram are in the ratio 3 : 2. Find the measure of each of the angles of the parallelogram.
Let the measures of two adjacent angles ∠A and ∠B be 3x and 2x, respectively in
parallelogram ABCD.
∠A + ∠B = 180°
⇒ 3x + 2x = 180°
⇒ 5x = 180°
⇒ x = 36°
We know that opposite sides of a parallelogram are equal.
∠A = ∠C = 3x = 3 × 36° = 108°
∠B = ∠D = 2x = 2 × 36° = 72°
Given a parallelogram ABCD. Complete each statement along with the definition or property used.
(i) AD = ……
(ii) ∠DCB = ……
(iii) OC = ……
(iv) m ∠DAB + m ∠CDA = ……
(i) AD = BC (Opposite sides of a parallelogram are equal)
(ii) ∠DCB = ∠DAB (Opposite angles of a parallelogram are equal)
(iii) OC = OA (Diagonals of a parallelogram are equal)
(iv) m ∠DAB + m ∠CDA = 180°
Consider the following parallelograms. Find the values of the unknown x, y, z
(i)
y = 100° (opposite angles of a parallelogram)
x + 100° = 180° (adjacent angles of a parallelogram)
⇒ x = 180° – 100° = 80°
x = z = 80° (opposite angles of a parallelogram)
∴, x = 80°, y = 100° and z = 80°
50° + x = 180° ⇒ x = 180° – 50° = 130° (adjacent angles of a parallelogram) x = y = 130° (opposite angles of a parallelogram)
x = z = 130° (corresponding angle)
(iii)
x = 90° (vertical opposite angles)
x + y + 30° = 180° (angle sum property of a triangle)
⇒ 90° + y + 30° = 180°
⇒ y = 180° – 120° = 60°
also, y = z = 60° (alternate angles)
(iv)
z = 80° (corresponding angle) z = y = 80° (alternate angles) x + y = 180° (adjacent angles)
⇒ x + 80° = 180° ⇒ x = 180° – 80° = 100°
(v)
x=28o
y = 112o z = 28o
Can a quadrilateral ABCD be a parallelogram if
(i) ∠D + ∠B = 180°?
(ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm?
(iii)∠A = 70° and ∠C = 65°?
(i) Yes, a quadrilateral ABCD can be a parallelogram if ∠D + ∠B = 180° but it should also fulfil some conditions, which are:
(a) The sum of the adjacent angles should be 180°.
(b) Opposite angles must be equal.
(ii) No, opposite sides should be of the same length. Here, AD ≠ BC
(iii) No, opposite angles should be of the same measures. ∠A ≠ ∠C
Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.
Let ABCD be a parallelogram.
Sum of adjacent angles of a parallelogram = 180°
∠A + ∠B = 180°
⇒ 2∠A = 180°
⇒ ∠A = 90°
also, 90° + ∠B = 180°
⇒ ∠B = 180° – 90° = 90°
∠A = ∠C = 90°
∠B = ∠D = 90
The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.
y = 40° (alternate interior angle)
∠P = 70° (alternate interior angle)
∠P = ∠H = 70° (opposite angles of a parallelogram)
z = ∠H – 40°= 70° – 40° = 30°
∠H + x = 180°
⇒ 70° + x = 180°
⇒ x = 180° – 70° = 110°
The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths are in cm)
(i) SG = NU and SN = GU (opposite sides of a parallelogram are equal) 3x = 18
x = 18/3
⇒ x =6
3y – 1 = 26
⇒ 3y = 26 + 1
⇒ y = 27/3=9
x = 6 and y = 9
(ii) 20 = y + 7 and 16 = x + y (diagonals of a parallelogram bisect each other) y + 7 = 20
⇒ y = 20 – 7 = 13 and,
x + y = 16
⇒ x + 13 = 16
⇒ x = 16 – 13 = 3
x = 3 and y = 13
In the above figure both RISK and CLUE are parallelograms. Find the value of x.
∠K + ∠R = 180° (adjacent angles of a parallelogram are supplementary)
⇒ 120° + ∠R = 180°
⇒ ∠R = 180° – 120° = 60°
also, ∠R = ∠SIL (corresponding angles)
⇒ ∠SIL = 60°
also, ∠ECR = ∠L = 70° (corresponding angles) x + 60° + 70° = 180° (angle sum of a triangle)
⇒ x + 130° = 180°
⇒ x = 180° – 130° = 50°
Explain how this figure is a trapezium. Which of its two sides are parallel? (Fig 3.32)
When a transversal line intersects two lines in such a way that the sum of the adjacent angles on the same side of transversal is 180°, then the lines are parallel to each other. Here, ∠M + ∠L = 100° + 80° = 180°
Thus, MN || LK
As the quadrilateral KLMN has one pair of parallel lines, it is a trapezium. MN and LK are parallel lines.
Find m∠C in Fig 3.33 if AB || DC.
m∠C + m∠B = 180° (angles on the same side of transversal)
⇒ m∠C + 120° = 180°
⇒ m∠C = 180°- 120° = 60°
Find the measure of ∠P and ∠S if SP || RQ ? in Fig 3.34. (If you find m∠R, is there more than one method to find m∠P?)
∠P + ∠Q = 180° (angles on the same side of transversal)
⇒ ∠P + 130° = 180°
⇒ ∠P = 180° – 130° = 50°
also, ∠R + ∠S = 180° (angles on the same side of transversal)
⇒ 90° + ∠S = 180°
⇒ ∠S = 180° – 90° = 90°
Thus, ∠P = 50° and ∠S = 90°
Yes, there are more than one method to find m∠P.
PQRS is a quadrilateral. Sum of measures of all angles is 360°.
Since, we know the measurement of ∠Q, ∠R and ∠S.
∠Q = 130°, ∠R = 90° and ∠S = 90°
∠P + 130° + 90° + 90° = 360°
⇒ ∠P + 310° = 360°
⇒ ∠P = 360° – 310° = 50°
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