NCERT Exemplar Problems Class 8 Mathematics Understanding Quadrilaterals and Practical Geometry

NCERT Exemplar Problems Class 8 Mathematics Chapter 5 Understanding Quadrilaterals and Practical Geometry

Multiple Choice Questions
Question. 1 If three angles of a quadrilateral are each equal to 75°, then, the fourth angle is(a) 150° (b) 135°
(c) 45° (d) 75°
Solution.

Question. 2 For which of the following, diagonals bisect each other?
(a) Square (b) Kite
(c) Trapezium (d) Quadrilateral
Solution. (a) We know that, the diagonals of a square bisect each other but the diagonals of kite, trapezium and quadrilateral do not bisect each other.

Question. 3 In which of the following figures, all angles are equal?
(a) Rectangle (b) Kite
(c) Trapezium (d) Rhombus
Solution. (a) In a rectangle, all angles are equal, i.e. all equal to 90°.

Question. 4 For which of the following figures, diagonals are perpendicular to each other?
(a) Parallelogram (b) Kite
(c) Trapezium (d) Rectangle
Solution. (b) The diagonals of a kite are perpendicular to each other.

Question. 5 For which of the following figures, diagonals are equal?
(a) Trapezium (b) Rhombus
(c) Parallelogram (d) Rectangle
Solution. (d) By the property of a rectangle, we know that its diagonals are equal.

Question. 6 Which of the following figures satisfy the following properties?
All sides are congruent
All angles are right angles.
Opposite sides are parallel.

Solution. (c) We know that all the properties mentioned above are related to square and we can observe that figure R resembles a square.

Question. 7 Which of the following figures satisfy the following property?Has two pairs of congruent adjacent sides.

Solution. (c) We know that, a kite has two pairs of congruent adjacent sides and we can observe that figure R resembles a kite.

Question. 8 Which of the following figures satisfy the following property?
Only one pair of sides are parallel.

Solution. (a) We know that, in a trapezium, only one pair of sides are parallel and we can observe that figure P resembles a trapezium.

Question. 9 Which of the following figures do not satisfy any of the following properties?
All sides are equal.
All angles are right angles.
Opposite sides are parallel.

Solution. (a) On observing the above figures, we conclude that the figure P does not satisfy any of the given properties.

Question. 10 Which of the following properties describe a trapezium?
(a) A pair of opposite sides is parallel
(b) The diagonals bisect each other
(c) The diagonals are perpendicular to each other
(d) The diagonals are equal
Solution. (a) We know that, in a trapezium, a pair of opposite sides are parallel.

Question. 11 Which of the following is a propefay of a parallelogram?
(a) Opposite sides are parallel
(b) The diagonals bisect each other at right angles
(c) The diagonals are perpendicular to each other
(d) All angles are equal
Solution. (a) We,know that, in a parallelogram, opposite sides are parallel.

Question. 12 What is the maximum number of obtuse angles that a quadrilateral can have?
(a) 1 (b) 2
(c) 3 (d) 4
Solution. (c) We know that, the sum of all the angles of a quadrilateral is 360°.
Also, an obtuse angle is more than 90° and less than 180°.
Thus, all the angles of a quadrilateral cannot be obtuse.
Hence, almost 3 angles can be obtuse.

Question. 13 How many non-overlapping triangles can we make in a-n-gon (polygon having n sides), by joining the vertices?
(a)n-1 (b)n-2
(c) n – 3 (d) n – 4
Solution. (b) The number of non-overlapping triangles in a n-gon = n – 2, i.e. 2 less than the number of sides.

Question. 14 What is the sum of all the angles of a pentagon?
(a) 180° (b) 360° (c) 540° (d) 720°
Solution. (c) We know that, the sum of angles of a polygon is (n – 2) x 180°, where n is the number of sides of the polygon.
In pentagon, n = 5
Sum of the angles = (n – 2) x 180° = (5 – 2) x 180°
= 3 x 180°= 540°

Question. 15 What is the sum of all angles of a hexagon?
(a) 180° (b) 360° (c) 540° (d) 720°
Solution. (d) Sum of all angles of a n-gon is (n – 2) x 180°.
In hexagon, n = 6, therefore the required sum = (6 – 2) x 180° = 4 x 180° = 720°

Question. 16 If two adjacent angles of a parallelogram are (5x – 5) and (10x + 35), then the ratio of these angles is
(a) 1 : 3 (b) 2 : 3 (c) 1 : 4 (d) 1 : 2
Solution.

Question. 17 A quadrilateral whose all sides are equal, opposite angles are equal and
the diagonals bisect each other at-right angles is a .
(a) rhombus (b) parallelogram (c) square (d) rectangle
Solution. (a) We know that, in rhombus, all sides are equal, opposite angles are equal and diagonals bisect each other at right angles.

Question. 18 A quadrilateral whose opposite sides and all the angles are equal is a
(a) rectangle (b) parallelogram (c) square (d) rhombus
Solution. (a) We know that, in a rectangle, opposite sides and all the angles are equal.

Question. 19 A quadrilateral whose all sides, diagonals and angles are equal is a
(a) square (b) trapezium (c) rectangle (d) rhombus
Solution. (a) These are the properties of a square, i.e. in a square, all sides, diagonals and angles are equal.

Question. 20 How many diagonals does a hexagon have?
(a) 9 (b) 8 (c) 2 (d) 6
Solution.

Question. 21 If the adjacent sides of a parallelogram are equal, then parallelogram is a
(a) rectangle (b) trapezium (c) rhombus (d) square
Solution. (c)We know that, in a parallelogram, opposite sides are equal.
But according to the question, adjacent sides are also equal.
Thus, the parallelogram in which all the sides are equal is known as rhombus.

Question. 22 If the diagonals of a quadrilateral are equal and bisect each other, then the quadrilateral is a
(a) rhombus (b) rectangle (c) square (d) parallelogram
Solution. (b) Since, diagonals are equal and bisect each other, therefore it will be a rectangle.

Question. 23 The sum of all exterior angles of a triangle is
(a) 180° (b) 360° (c) 540° (d) 720°
Solution. (b) We know that the sum of exterior angles, taken in order of any polygon is 360° and triangle is also a polygon.
Hence, the sum of all exterior angles of a triangle is 360°.

Question. 24 Which of the following is an equiangular and equilateral polygon?
(a) Square (b) Rectangle (c) Rhombus (d) Right triangle
Solution. (a) In a square, all the sides and all the angles are equal.
Hence, square is an equiangular and equilateral polygon.

Question. 25 Which one has all the properties of a kite and a parallelogram?
(a) Trapezium (b) Rhombus (c) Rectangle (d) Parallelogram
Solution. (b) In a kite
Two pairs of equal sides.
Diagonals bisect at 90°.
One pair of opposite angles are equal.
In a parallelogram Opposite sides are equal.
Opposite angles are equal.
Diagonals bisect each other.
So, from the given options, all these properties are satisfied by rhombus.

Question. 26 The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. The smallest angle is
(a) 72° (b) 144° (c) 36° (d) 18°
Solution.

Question. 27 In the trapezium ABCD, the measure of \(\angle D\) is
(a) 55° (b) 115° (c)135° (d) 125°

Solution.

Question. 28 A quadrilateral has three acute angles. If each measures 80°, then the measure of the fourth angle is
(a) 150° (b) 120° (c) 105° (d) 140°
Solution.

Question. 29 The number of sides of a regular polygon where each exterior angle has a measure of 45° is
(a) 8 (b) 10 (c) 4 (d) 6
Solution.

Question. 30 In a parallelogram PQRS, if \(\angle P\) = 60°, then other three angles are
(a) 45°, 135°, 120° (b) 60°, 120°, 120°
(c) 60°, 135°, 135° (d) 45°, 135°, 135°
Solution.

Question. 31 If two adjacent angles of a parallelogram are in the ratio 2 : 3, then the measure of angles are
(a) 72°, 108° (b) 36°, 54° (c) 80°, 120° (d) 96°, 144°
Solution. (a) Let the angles be 2x and 3x.
Then, 2x + 3x = 180° [ adjacent angles of a parallelogram are supplementary]
=> 5x = 180°
=> x = 36°
Hence, the measures of angles are 2x = 2 x 36°= 72° and 3x = 3×36°= 108°

Question. 32 IfPQRS is a parallelogram then \(\angle P\) – \(\angle R\) is equal to
(a) 60° (b) 90° (c) 80° (d) 0°
Solution. (d) Since, in a parallelogram, opposite angles are equal. Therefore, \(\angle P\) – \(\angle R\) = 0, as \(\angle P\) and \(\angle R\) are opposite angles.

Question. 33 The sum of adjacent angles of a parallelogram is
(a) 180° (b) 120° (c) 360° (d) 90°
Solution. (a) By property of the parallelogram, we know that, the sum of adjacent angles of a parallelogram is 180°.

Question. 34 The angle between the two altitudes of a parallelogram through the same vertex of an obtuse angle of the parallelogram is 30°. The measure of the obtuse angle is
(a) 100° (b) 150° (c) 105° (d) 120°
Solution.

Question. 35 In the given figure, ABCD and BDCE are parallelograms with common base DC. If \(BC\bot BD\), then \(\angle BEC\) is equal to
(a) 60° (b) 30° (c) 150° (d) 120°

Solution.

Question. 36 Length of one of the diagonals of a rectangle whose sides are 10 cm and 24 cm is
(a) 25 cm (b) 20 cm (c) 26 cm (d) 3.5 cm
Solution.

Question. 37 If the adjacent angles of a parallelogram are equal, then the parallelogram is a (a) rectangle (b) trapezium (c) rhombus (d) None of these
Solution. (a) We know that, the adjacent angles of a parallelogram are supplementary, i.e. their sum equals 180° and given that both the angles are same. Therefore, each angle will be of measure 90°. .
Hence, the parallelogram is a rectangle.

Question. 38 Which of the following can be four interior angles of a quadrilateral?
(a) 140°, 40°, 20°, 160° (b) 270°, 150°, 30°, 20°
(c) 40°, 70°, 90°, 60°    (d) 110°, 40°, 30°, 180°
Solution. (a) We know that, the sum of interior angles of a quadrilateral is 360°.
Thus, the angles in option (a) can be four interior angles of a quadrilateral as their sum is 360°.

Question. 39 The sum of angles of a concave quadrilateral is
(a) more than 360° (b) less than 360°
(c) equal to 360° (d) twice of 360°
Solution. (c) We know that, the sum of interior angles of any polygon (convex or concave) having n sides is(n -2) x 180°.
.-.The sum of angles of a concave quadrilateral is (4 – 2) x 180°, i.e. 360°

Question. 40 Which of the following can never be the measure of exterior angle of a regular polygon? (a) 22° (b) 36° (c)45° (d) 30°
Solution. (a) Since, we know that, the sum of measures of exterior angles of a polygon is 360°, i.e. measure of each exterior angle =360°/n ,where n is the number of sides/angles.
Thus, measure of each exterior angle will always divide 360° completely.
Hence, 22° can never be the measure of exterior angle of a regular polygon.

Question. 41 In the figure, BEST is a rhombus, then the value of y – x is
(a) 40° (b) 50° (c) 20° (d) 10°

Solution.

Question. 42 The closed curve which is also a polygon, is

Solution. (a) Figure (a) is polygon as no two line segments intersect each other.

Question. 43 Which of the following is not true for an exterior angle of a regular polygon with n sides?

Solution. (d) We know that, (a) and (b) are the formulae to find the measure of each exterior angle, when number of sides and measure of an interior angle respectively are given and (c) is the formula to find number of sides of polygon when exterior angle is given.
Hence, the formula given in option (d) is not true for an exterior angle of a regular polygon with n sides.

Question. 44 PQRS is a square. PR and SQ intersect at 0. Then, \(\angle POQ\) is a (a) right angle (b) straight angle (c) reflex angle (d) complete angle
Solution.

Question. 45 Two adjacent angles of a parallelogram are in the ratio 1 : 5. Then, all the angles of the parallelogram are
(a) 30°, 150°, 30°, 150° (b) 85°, 95°, 85°, 95° .
(c) 45°, 135°, 45°, 135° (d) 30°, 180°, 30°, 180°
Solution. (a) Let the adjacent angles of a parallelogram be x and 5x, respectively.
Then, x + 5x = 180° [ adjacent angles of a parallelogram are supplementary] => 6x = 180°
=> x = 30°
The adjacent angles are 30° and 150°.
Hence, the angles are 30°, 150°, 30°, 150°

Question. 46 A parallelogram PQRS is constructed with sides QR = 6 cm, PQ = 4 cm and \(\angle PQR\) = 90°. Then, PQRS is a
(a) square (b) rectangle (c) rhombus  (d) trapezium
Solution. (b) We know that, if in a parallelogram one angle is of 90°, then all angles will be of 90° and a parallelogram with all angles equal to 90° is called a rectangle.

Question. 47 The angles P, Q, R and 5 of a quadrilateral are in the ratio  1:3 :7:9. Then, PQRS is a
(a) parallelogram  (b) trapezium with PQ \\ RS
(c) trapezium with  QR \\PS (d) kite
Solution.

Question. 48 PQRS is a trapezium in which PQ || SR and ZP = 130°, \(\angle Q\) = 110°. Then, \(\angle R\) is equal to.
(a) 70° (b) 50° (c)65° (d) 55°
Solution.

Question. 49 The number of sides of a regular polygon whose each interior angle is of 135° is (a) 6  (b) 7 (c) 8 (d) 9
Solution.

Question. 50 If a diagonal of a quadrilateral bisects both the angles, then it is a
(a) kite (b) parallelogram  (c) rhombus (d) rectangle
Solution. (c) If a diagonal of a quadrilateral bisects both the angles, then it is a rhombus.

Question. 51 To construct a unique parallelogram, the minimum number of measurements required is (a) 2 (b) 3 (c) 4 (d) 5
Solution. (b) We know that, in a parallelogram, opposite sides are equal and parallel. Also,
opposite angles are equal.
So, to construct a parallelogram uniquely, we require the measure of any two non-parallel sides and the measure of an angle.
Hence, the minimum number of measurements required to draw a unique parallelogram is 3.

Question. 52 To construct a unique rectangle, the minimum number of measurements required is (a) 4 (b) 3 (0 2 (d) 1
Solution. (c) Since, in a rectangle, opposite sides are equal and parallel, so we need the measurement of only two adjacent sides, i.e. length and breadth. Also, each angle measures 90°.
Hence, we require only two measurements to construct a unique rectangle.

Fill in the Blanks
In questions 53 to 91, fill in the blanks to make the statements true.
Question. 53 In quadrilateral HOPE, the pairs of opposite sides are————–.
Solution.

Question. 54 In quadrilateral ROPE, the pairs of adjacent angles are—————-.
Solution .

Question. 55 In quadrilateral WXYZ, the pairs of opposite angles are————–.
Solution.

Question . 56 The diagonals of the quadrilateral DEFG are———–and————–.
Solution.

Question. 57 The sum of all———— of a quadrilateral is 360°.
Solution. angles
We know that, the sum of all angles of a quadrilateral is 360°.

Question. 58 The measure of each exterior angle of a regular pentagon is————— .
Solution.

Question. 59 Sum of the angles of a hexagon is———————-.
Solution.

Question. 60 The measure of each exterior angle of a regular polygon of 18 sides is———.
Solution.

Question. 61 The number of sides of a regular polygon, where each exterior angle has a measure of 36°, is—————-.
Solution.

Question. 62

Solution. concave polygon
As one interior angle is of greater than 180°.

Question. 63 A quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure is—————–.
Solution. kite
By the property of a kite, we know that, it has two opposite angles of equal measure.

Question. 64 The measure of each angle of a regular pentagon is————–.
Solution.

Question. 65 The name of three-sided regular polygon is—————-.
Solution. equilateral triangle, as polygon is regular, i.e. length of each side is same.

Question. 66 The number of diagonals in a hexagon is—————-.
Solution.

Question. 67 A polygon is a simple closed curve made up of only————.
Solution. line segments ,
Since a simple closed curve made up of only line segments is called a polygon.

Question. 68 A regular polygon is a polygon whose all sides are equal and all———are equal.
Solution. angles
In a regular polygon, all sides are equal and all angles are equal.

Question. 69 The sum of interior angles of a polygon of n sides is———- right angles.
Solution.

Question. 70 The sum of all exterior angles of a polygon is————.
Solution. 360°
As the sum of all exterior angles of a polygon is 360°.

Question. 71 ————-is a regular quadrilateral.
Solution. Square
Since in square, all the sides are of equal length and all angles are equal.

Question. 72 A quadrilateral in which a pair of opposite sides is parallel is————-.
Solution. trapezium
We know that, in a trapezium, one pair of sides is parallel.

Question. 73 If all sides of a quadrilateral are equal, it is a————–.
Solution. rhombus or square
As in both the quadrilaterals all sides are of equal length.

Question. 74 In a rhombus, diagonals intersect at———– angles.
Solution. right
The diagonals of a rhombus intersect at right angles.

Question. 75 ———measurements can determine a quadrilateral uniquely.
Solution. 5
To construct a unique quadrilateral, we require 5 measurements, i.e. four sides and one angle or three sides and two included angles or two adjacent sides and three angles are given.

Question. 76 A quadrilateral can be constructed uniquely, if its three sides and———–angles are given.
Solution. two included
We cap determine a quadrilateral uniquely, if three sides and two included angles are given.

Question. 77 A rhombus is a parallelogram in which————sides are equal.
Solution. all
As length of each side is same in a rhombus.

Question. 78 The measure of——– angle of concave quadrilateral is more than 180°.
Solution. one
Concave polygon is a polygon in which at least one interior angle is more than 180°.

Question. 79 A diagonal of a quadrilateral is a line segment that joins two——– vertices of the quadrilateral.
Solution. opposite
Since the line segment connecting two opposite vertices is called diagonal.

Question. 80 The number of sides in a regular polygon having measure of an exterior angle as 72° is————— .
Solution. 5
We know that,the sum of exterior angles of any polygon is 360°.

Question. 81 If the diagonals of a quadrilateral bisect each other, it is a————.
Solution. parallelogram
Since in a parallelogram, the diagonals bisect each other.

Question. 82 The adjacent sides of a parallelogram are 5 cm and 9 cm. Its perimeter is—–.
Solution. 28 cm
Perimeter of a parallelogram = 2 (Sum of lengths of adjacent sides)
=2(5+ 9) = 2 x 14=28cm

Question. 83 A nonagon has————sides.
Solution. 9
Nonagon is a polygon which has 9 sides.

Question. 84 Diagonals of a rectangle are————.
Solution. equal
We know that, in a rectangle, both the diagonals are of equal length.

Question. 85 A polygon having 10 sides is known as————.
Solution. decagon
A polygon with 10 sides is called decagon.

Question. 86 A rectangle whose adjacent sides are equal becomes a ————.
Solution. square
If in a rectangle, adjacent sides are equal, then it is called a square.

Question. 87 If one diagonal of a rectangle is 6 cm long, length of the other diagonal is—–.
Solution. 6 cm
Since both the diagonals of a rectangle are equal. Therefore, length of other diagonal is also 6 cm.

Question. 88 Adjacent angles of a parallelogram are————.
Solution. supplementary
By property of a parallelogram, we know that, the adjacent angles of a parallelogram are supplementary.

Question. 89 If only one diagonal of a quadrilateral bisects the other, then the quadrilateral is known as————.
Solution. kite
This is a property of kite, i.e. only one diagonal bisects the other.

Question. 90 In trapezium ABCD with AB || CD, if \( \angle A\)= 100°, then \( \angle D\) =————.
Solution.

Question. 91 The polygon in which sum of all exterior angles is equal to the sum of interior angles is called————.
Solution. quadrilateral
We know that, the sum of exterior angles of a polygon is 360° and in a quadrilateral, sum of interior angles is also 360°. Therefore, a quadrilateral is a polygon in which the sum of both interior and exterior angles are equal.

True/False
In questions 92 to 131, state whether the statements are True or False.
Question. 92 All angles of a trapezium are equal.
Solution. False
As all angles of a trapezium are not equal.

Question. 93 All squares are rectangles.
Solution. True
Since squares possess all the properties of rectangles. Therefore, we can say that, all squares are rectangles but vice-versa is not true.

Question. 94 All kites are squares.
Solution. False
As kites do not satisfy all the properties of a square.
e.g. In square, all the angles are of 90° but in kite, it is not the case.

Question. 95 All rectangles are parallelograms.
Solution. True
Since rectangles satisfy all ”the”properties” of parallelograms. Therefore, we can say that, all rectangles are parallelograms but vice-versa is not true.

Question. 96 All rhombuses are square.
Solution. False
As in a rhombus, each angle is not a right angle, so rhombuses are not squares.

Question. 97 Sum of all the angles of a quadrilateral is 180°.
Solution. False
Since sum of all the angles of a quadrilateral is 360°.

Question. 98 A quadrilateral has two diagonals.
Solution. True
A quadrilateral has two diagonals.

Question. 99 Triangle is a polygon whose sum of exterior angles is double the sum of interior angles.
Solution. True
As the sum of interior angles of a triangle is 180° and the sum of exterior angles is 360°, i.e. double the sum of interior angles.

Question. 100

Solution. False
Because it is not a simple closed curve as it intersects with itself more than once.

Question. 101 A kite is not a convex quadrilateral.
Solution. False
A kite is a convex quadrilateral as the line segment joining any two opposite vertices inside it, lies completely inside it.

Question. 102 The sum of interior angles and the sum of exterior angles taken in an order are equal in case of quadrilaterals only.
Solution. True
Since the sum of interior angles as well as of exterior angles of a quadrilateral are 360°.

Question. 103 If the sum of interior angles is double the sum of exterior angles taken in an order of a polygon, then it is a hexagon.
Solution. True
Since the sum of exterior angles of a hexagon is 360° and the sum of interior angles of a hexagon is 720°, i.e. double the sum of exterior angles.

Question. 104 A polygon is regular, if all of its sides are equal.
Solution. False
By definition of a regular polygon, we know that, a polygon is regular, if all sides and all angles are equal.

Question. 105 Rectangle is a regular quadrilateral.
Solution. False
As its all sides are not equal.

Question. 106 If diagonals of a quadrilateral are equal, it must be a rectangle.
Solution. True
If diagonals are equal, then it is definitely a rectangle. –

Question. 107 If opposite angles of a quadrilateral are equal, it must be a parallelogram.
Solution. True
If opposite angles are equal, it has to be a parallelogram.

Question. 108 The interior angles of a triangle are in the ratio 1:2:3, then the ratio of its exterior angles is 3 : 2 : 1.
Solution.

Question. 109

Solution. False
As it has 6 sides, therefore it is a concave hexagon.

Question. 110 Diagonals of a rhombus are equal and perpendicular to each other.
Solution. False
As diagonals of a rhombus are perpendicular to each other but not equal.

Question. 111 Diagonals of a rectangle are equal.
Solution. True
The diagonals of a rectangle are equal.

Question. 112 Diagonals of rectangle bisect each other at right angles.
Solution. False
Diagonals of a rectangle does not bisect each other.

Question. 113 Every kite is a parallelogram.
Solution. False
Kite is not a parallelogram as its opposite sides are not equal and parallel.

Question. 114 Every trapezium is a parallelogram.
Solution. False
Since in a trapezium, only one pair of sides is parallel.

Question. 115 Every parallelogram is a rectangle.
Solution. False
As in a parallelogram, all angles are not right angles, while in a rectangle, all angles are equal and are right angles.

Question. 116 Every trapezium is a rectangle.
Solution. False
Since in a rectangle, opposite sides are equal and parallel but in a trapezium, it is not so.

Question. 117 Every rectangle is a trapezium.
Solution. True
As a rectangle satisfies all the properties of a trapezium. So, we can say that, every rectangle is a trapezium but vice-versa is not true.

Question. 118 Every square is a rhombus.
Solution. True
As a square possesses all the properties of a rhombus. So, we can say that, every square is a rhombus but vice-versa is not true.

Question. 119 Every square is a parallelogram.
Solution. True
Every square is also a parallelogram as it has all the properties of a parallelogram but vice-versa is not true.

Question. 120 Every square is a trapezium.
Solution. True
As a square has all the properties of a trapezium. So, we can say that, every square is a trapezium but vice-versa is not true.

Question. 121 Every rhombus is a trapezium.
Solution. True
Since a rhombus satisfies all the properties of a trapezium. So, we can say that, every rhombus is a trapezium but vice-versa is not true.

Question. 122 A quadrilateral can be drawn if only measures of four sides are given.
Solution. False
As we require at least five measurements to determine a quadrilateral uniquely.

Question. 123 A quadrilateral can have all four angles as obtuse.
Solution. False
If all angles will be obtuse, then their sum will exceed 360°. This is not possible in case of a quadrilateral.

Question. 124 A quadrilateral can be drawn, if all four sides and one diagonal is known.
Solution. True
A quadrilateral can be constructed uniquely, if four sides and one diagonal is known.

Question. 125 A quadrilateral can be drawn, when all the four angles and one side is given.
Solution. False
We cannot draw a unique-quadrilateral, if four angles and one side is known.

Question. 126 A quadrilateral can be drawn, if all four sides and one angle is known.
Solution. True
A quadrilateral can be drawn, if all four sides and one angle is known.

Question. 127 A quadrilateral can be drawn, if three sides and two diagonals are given.
Solution. True
A quadrilateral can be drawn, if three sides and two diagonals are given.

Question. 128 If diagonals of a quadrilateral bisect each other, it must be a parallelogram.
Solution. True
It is the property of a parallelogram.

Question. 129 A quadrilateral can be constructed uniquely, if three angles and any two included sides are given.
Solution. True
We can construct a unique quadrilateral with given three angles given and two included sides.

Question. 130 A parallelogram can be constructed uniquely, if both diagonals and the angle between them is given.
Solution. True
We can draw a unique parallelogram, if both diagonals and the angle between them is given.

Question. 131 A rhombus can be constructed uniquely, if both diagonals are given.
Solution. True
A rhombus can be constructed uniquely, if both diagonals are given.

Question. 132 The diagonals of a rhombus are 8 cm and 15 cm. Find its side.
Solution.

Question. 133 Two adjacent angles of a parallelogram are in the ratio 1 : 3. Find its angles.
Solution. Let the adjacent angles of a parallelogram be x and 8c.
Then, we have x + (3 x) = 180° [adjacent angles of parallelogram are supplementary]
=> 4 x = 180°
=> x = 45°
Thus, the angles are 45°, 135°.
Hence, the angles are 45°, 135, 45°, 135°. [ opposite angles in a parallelogram are equal]

Question. 134 Of the four quadrilaterals – square, rectangle, rhombus and trapezium-one is somewhat different from the others because of its design. Find it and give justification.
Solution. In square, rectangle and rhombus, opposite sides are parallel and equal. Also, opposite angles are equal, i.e. they all are parallelograms.
But in trapezium, there is only one pair of parallel sides, i.e. it is not a parallelogram. Therefore, trapezium has different design.

Question. 135 In a rectangle ABCD, AB = 25 cm and BC = 15 cm. In what ratio, does the bisector of \(\angle C\) divide AB?
Solution.

Question. 136 PQRS is a rectangle. The perpendicular ST from S on PR divides \(\angle S\) in the ratio 2 : 3. Find \(\angle TPQ\).
Solution.

Question. 137 A photo frame is in the shape of a quadrilateral, with one diagonal longer than the other. Is it a rectangle? Why or why not?
Solution. No, it cannot be a rectangle, as in rectangle, both the diagonals are of equal lengths.

Question. 138 The adjacent angles of a parallelogram are (2x – 4)° and (3x – 1)°. Find the measures of all angles of the parallelogram.
Solution. Since, the adjacent angles of a parallelogram are supplementary.
(2 x – 4)° + (3* – 1)° = 180°

Question. 139 The point of intersection of diagonals of a quadrilateral divides one diagonal in the ratio 1: 2. Can it be a parallelogram? Why or why not?
Solution. No, it can never be a parallelogram, as the diagonals of a parallelogram intersect each other in the ratio 1 : 1.

Question. 140 The ratio between exterior angle and interior angle of a regular polygon is 1 : 5. Find the number of sides of the polygon.
Solution.

Question. 141 Two sticks each of length 5 cm are crossing each other such that they bisect each other. What shape is formed by joining their end points? Give reason.
Solution. Sticks can be taken as the diagonals of a quadrilateral.
Now, since they are bisecting each other, therefore the shape formed by joining their end points will be a parallelogram.
Hence, it may be a rectangle or a square depending on the angle between the sticks.

Question. 142 Two sticks each of length 7 cm are crossing each other such that they bisect each other at right angles. What shape is formed by joining their end points? Give reason.
Solution. Sticks can be treated as the diagonals of a quadrilateral.
Now, since the diagonals (sticks) are bisecting each other at right angles, therefore the shape formed by joining their end points will be a rhombus.

Question. 143 A playground in the town is in the form of a kite. The perimeter is 106 m. If one of its sides is 23 m, what are the lengths of other three sides?
Solution. Let the length of other non-consecutive side be x cm.
Then, we have, perimeter of playground = 23 + 23+ x + x
=> 106 = 2 (23+ x)
=>46 + 2x = 106 2x = 106 – 46
=>2x = 60
=>x = 30 m
Hence, the lengths of other three sides are 23m, 30m and 30m. As a kite has two pairs of equal consecutive sides.

Question. 144 In rectangle READ , find \(\angle EAR\), \(\angle RAD\) and \(\angle ROD\).

Solution.

Question. 145 In rectangle PAIR, find \(\angle ARI\), ZRMI and \(\angle PMA\).

Solution.

Question. 146 In parallelogram ABCD, find \(\angle B\), \(\angle C\) and \(\angle D\).

Solution.

Question. 147 In parallelogram PQRS, 0 is the mid-point of SQ. Find \(\angle S\), \(\angle R\), PQ, QR and diagonal PR.

Solution.

Question. 148 In rhombus BEAM, find \(\angle AME\) and \(\angle AEM\).

Solution.

Question. 149 In parallelogram FIST, find \(\angle SFT\), \(\angle OST\) and \(\angle STO\).

Solution.

Question. 150 In the given parallelogram YOUR, \(\angle RUO\)= 120° and 0Y is extended to points, such that \(\angle SRY\) = 50°. Find \(\angle YSR\).

Solution.

Question.151 In kite WEAR, \(\angle WEA\) = 70° and \(\angle ARW\) = 80°. Find the remaining two angles.

Solution.

Question.152

Solution.

Question.153 In parallelogram LOST, SNLOL and \( SM\bot LT\). Find \(\angle STM\), \(\angle SON\) and \(\angle NSM\).

Solution.

Question. 154 In trapezium HARE, EP and RP are bisectors of \(\angle E\) and \(\angle R\), respectively. Find \(\angle HAR\) and \(\angle EHA\).

Solution.

Question. 155 In parallelogram MODE, the bisectors of \(\angle M\) and \(\angle O\) meet at Q. Find the measure of \(\angle MQO\).
Solution.

Question. 156 A playground is in the form of a rectangle ATEF. Two players are standing at the points F and B, where EF =EB. Find the values of x and y.

Solution.

Question. 157 In the following figure of a ship, ABDH and CEFG are two parallelograms. Find the value of x.

Solution.

Question. 158 A rangoli has been drawn on the floor of a house. ABCD and PQRS both are in the shape of a rhombus. Find the radius of semi-circle drawn on each side of rhombus ABCD.

Solution.

Question. 159 ABCDE is a regular pentagon. The bisector of angle A meets the sides CD at M. Find \(\angle AMC\)

Solution.

Question. 160 Quadrilateral EFGH is a rectangle in which J is the point of intersection of the diagonals. Find the value of x, if JF = 8x + 4 and EG = 24 x – 8.
Solution.

Question. 161 Find the values of x and y in the following parallelogram.

Solution.

Question. 162 Find the values of x and y in the following kite.

Solution.

Question. 163 Find the value of x in the trapezium ABCD given below.

Solution.

Question. 164 Two angles of a quadrilateral are each of measure 75° and the other two angles are equal. What is the measure of these two angles? Name the possible figures so formed.
Solution.

Question. 165 In a quadrilateral PQRS, \(\angle P\) = 50°, \(\angle Q\) = 50°, \(\angle R\) = 60°. Find \(\angle S\). Is this quadrilateral convex or concave?
Solution.

Question. 166 Both the pairs of opposite angles of a quadrilateral are equal and supplementary. Find the measure of each angle.
Solution.

Question. 167 Find the measure of each angle of a regular octagon.
Solution.

Question. 168 Find the measure of an exterior angle of a regular pentagon and an exterior angle of a regular decagon. What is the ratio between these two angles?
Solution.

Question. 169 In the figure, find the value of x.

Solution.

Question. 170 Three angles of a quadrilateral are equal. Fourth angle is of measure 120°. What is the measure of equal angles?
Solution.

Question. 171 In a quadrilateral HOPE, PS and ES are bisectors of \(\angle P\) and \(\angle E\) respectively. Give reason.
Solution. Data insufficient.

Question. 172 ABCD is a parallelogram. Find the values of x, y and z.

Solution.

Question. 173 Diagonals of a quadrilateral are perpendicular to each other. Is such a quadrilateral always a rhombus? Give a figure to justify your answer.
Solution.

Question. 174 ABCD is a trapezium such that AB || CD, \(\angle A\): \(\angle D\) = 2:1, \(\angle B\) : \(\angle C\) = 7:5. Find the angles of the trapezium.
Solution.

Question. 175 A line / is parallel to Line m and a-transversal p intersects them at X, Y respectively. Bisectors of interior angles at X and Y intersect at P and Q. Is PXQY a rectangle? Give reason.
Solution.

Question. 176 ABCD is a parallelogram. The bisector of angle A intersects CD at X and bisector of angle C intersects AB at Y. Is AXCY a parallelogram? Give reason.
Solution.

Question. 177 A diagonal of a parallelogram bisects an angle. Will it also bisect the other angle? Give reason.
Solution. Consider a parallelogram ABCD.
Given, \(\angle 1\) = \(\angle 2\)

Question. 178 The angle between the two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is 45°. Find the angles of the parallelogram.
Solution. Let ABCD be a parallelogram, where BE and BF are the perpendiculars through the vertex B to the sides DC and AD, respectively.

Question. 179 ABCD is a rhombus such that the perpendicular bisector of AB passes through D. Find the angles of the rhombus.[Hint Join BD. Then, AABD is equilateral.]
Solution. Let ABCD be a rhombus in which DE is perpendicular bisector of AB.

Question. 180 ABCD is a parallelogram. Point P and Q are taken on the sides AB and AD, respectively and 4he parallelogram PRQA is formed. If \(\angle C\)= 45°, find \(\angle R\).
Solution.

Question. 181 In parallelogram ABCD, the angle bisector of \(\angle A\) bisects BC. Will angle bisector of B also bisect AD? Give reason.
Solution. Given, ABCD is a parallelogram, bisector of \(\angle A\), bisects BC at F, i.e. \(\angle 1\) = \(\angle 2\),CF = FB Draw FE || BA.

Question. 182 A regular pentagon ABCDE and a square ABFG are formed on opposite sides of AB. Find \(\angle BCF\)?
Solution.

Question. 183 Find maximum number of acute angles which a convex quadrilateral, a pentagon and a hexagon can have. Observe the pattern and generalise the result for any polygon.
Solution. If an angle is acute, then the corresponding exterior angle is greater than 90°. Now, suppose a convex polygon has four or more acute angles. Since, the polygon is convex, all the exterior angles are positive, so the sum of the exterior angle is at least the sum of the interior angles. Now, supplementary of the four acute angles, which is greater than 4 x 90° = 360°
However, this is impossible. Since, the sum of exterior angle of a polygon must equal to 360° and cannot be greater than it. It follows that the maximum number of acute angle in convex polygon is 3.

Question. 184 In the following figure, FD || BC || AE and AC || ED. Find the value of x.

Solution.

Question. 185 In the following figure, AB || DC and AD = BC. Find the value of x.

Solution.

Question. 186 Construct a trapezium ABCD in which AB || DC, \(\angle A\) = 105°, AD = 3 cm, AB = 4 cm and CD = 8 cm.
Solution.

Question. 187 Construct a parallelogram ABCD in which AB =4 cm, BC = 5cm and \(\angle B\) = 60°.
Solution.

Question. 188 Construct a rhombus whose side is 5 cm and one angle is of 60°
Solution.

Question. 189 Construct a rectangle whose one side is 3 cm and a diagonal is equal to 5 cm.
Solution.

Question. 190 Construct a square of side 4 cm.
Solution.

Question. 191 Construct a rhombus CLUE in which CL = 7.5 cm and LE = 6 cm.
Solution.

Question. 192 Construct a quadrilateral BEAR in which BE = 6 cm, EA = 7 cm, RB = RE = 5 cm and BA = 9 cm. Measure its fourth side.
Solution.

Question. 193 Construct a parallelogram POUR in which PO = 5.5 cm, OU = 7.2 cm and \(\angle O\) = 70°.
Solution.

Question. 194 Draw a circle of radius 3 cm and draw its diameter and label it as AC. Construct its perpendicular bisector and let it intersect the circle at B and D. What type of quadrilateral is ABCD? Justify your answer.
Solution.

Question. 195 Construct a parallelogram HOME with HO = 6 cm, HE = 4 cm and OE = 3 cm.
Solution.

Question. 196 Is it possible to construct a quadrilateral ABCD in which AB = 3 cm, BC = 4 cm, CD = 5.4 cm, DA = 5.9 cm and diagonal AC = 8 cm? If not, why?
Solution. No,
Given measures are AS = 3 cm, SC = 4 cm,CD = 5.4 cm,
DA = 59cmand AC = 8cm
Here, we observe that AS + SC = 3 + 4 = 7 cm and AC = 8 cm
i.e. the sum of two sides of a triangle is less than the third side, which is absurd.
Hence, we cannot construct such a quadrilateral.

Question. 197 Is it possible to construct a quadrilateral ROAM in which RO = 4 cm, OA = 5 cm, \(\angle O\) = 120°,\(\angle R\) = 105° and \(\angle A\) = 135°? If not, why?
Solution.

Question. 198 Construct a square in which each diagonal is 5 cm long.
Solution.

Question. 199 Construct a quadrilateral NEWS in which NE = 7 cm, EW = 6 cm, \(\angle N\) = 60°, \(\angle E\)= 110° and \(\angle S\) = 85°
Solution.

Question. 200 Construct a parallelogram when one of its side is 4 cm and its two diagonals are 5.6 cm and 7 cm. Measure the other side.
Solution.

Question. 201 Find the measure of each angle of a regular polygon of 20 sides?
Solution.

Question. 202 Construct a trapezium RISK in which RI || KS, RI = 7 cm, IS = 5 cm, RK = 6.5 cm and \(\angle I\) = 60°.
Solution.

Question. 203 Construct a trapezium ABCD, where AB|| CD, AD = BC = 3.2 cm, AB = 6.4 cm and CD = 9.6 cm. Measure \(\angle B\) and \(\angle A\)

[Hint Difference of two parallel sides gives an equilateral triangle.]
Solution.

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