NCERT Exemplar Solutions Class 8 Maths Chapter 5 Understanding Quadrilaterals & Practical Geometry

# NCERT Exemplar Solutions Class 8 Maths Chapter 5 Understanding Quadrilaterals & Practical Geometry

## NCERT Exemplar Solutions for Class 8 Maths Chapter 5

Multiple Choice Questions

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Question. For which of the following, diagonals bisect each other?

(a) Square

(b) Kite

(c) Trapezium

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. 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. 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. 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. 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. 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. Which of the following figures satisfy the following property?

Only one pair of sides are parallel.

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

Question. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. If PQRS is a parallelogram then PR is equal to

(a) 60°

(b) 90°

(c) 80°

(d) 0°

Solution. (d) Since, in a parallelogram, opposite angles are equal. Therefore, PR = 0, as P and R are opposite angles.

Question. 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. 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. 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. 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. 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. 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. A parallelogram PQRS is constructed with sides QR = 6 cm, PQ = 4 cm and 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. 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. 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. 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

Question. 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. The name of three-sided regular polygon is—————-.

Solution. equilateral triangle, as polygon is regular, i.e. length of each side is same.

Question. 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. 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. The sum of all exterior angles of a polygon is————.

Solution. 360°. As the sum of all exterior angles of a polygon is 360°.

Solution. Square, Since in square, all the sides are of equal length and all angles are equal.

Question. 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. 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. In a rhombus, diagonals intersect at———– angles.

Solution. Right, the diagonals of a rhombus intersect at right angles.

Question. ———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. 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. A rhombus is a parallelogram in which————sides are equal.

Solution. all

As length of each side is same in a rhombus.

Question. 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. 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. 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. If the diagonals of a quadrilateral bisect each other, it is a————.

Solution. parallelogram, Since in a parallelogram, the diagonals bisect each other.

Question. 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. A nonagon has————sides.

Solution. 9

Nonagon is a polygon which has 9 sides.

Question. Diagonals of a rectangle are————.

Solution. equal

We know that, in a rectangle, both the diagonals are of equal length.

Question. A polygon having 10 sides is known as————.

Solution. decagon

A polygon with 10 sides is called decagon.

Question. 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. 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. 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. 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. The polygon in which sum of all exterior angles is equal to the sum of interior angles is called————.

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

Question. All angles of a trapezium are equal.

Solution. False

As all angles of a trapezium are not equal.

Question. 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. 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. 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. All rhombuses are square.

Solution. False

As in a rhombus, each angle is not a right angle, so rhombuses are not squares.

Question. Sum of all the angles of a quadrilateral is 180°.

Solution. False

Since sum of all the angles of a quadrilateral is 360°.

Question. A quadrilateral has two diagonals.

Solution. True

Question. 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. 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. 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. 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. 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. Rectangle is a regular quadrilateral.

Solution. False

As its all sides are not equal.

Question. 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. 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. 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. Diagonals of a rectangle are equal.

Solution. True

The diagonals of a rectangle are equal.

Question. Diagonals of rectangle bisect each other at right angles.

Solution. False

Diagonals of a rectangle does not bisect each other.

Question. Every kite is a parallelogram.

Solution. False

Kite is not a parallelogram as its opposite sides are not equal and parallel.

Question. Every trapezium is a parallelogram.

Solution. False

Since in a trapezium, only one pair of sides is parallel.

Question. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.

## Class 8 Maths Chapter 5 FAQs

### Where can I get the accurate solution for the NCERT Solution of this chapter?

At Infinity Learn you can get a PDF of accurate solutions to this chapter. The NCERT Textbook Solutions for this chapter have been formulated by mathematics experts at Infinity Learn. All these solutions are according to the new pattern of CBSE, so the students can be ready for the exams.

### Is it necessary to solve each problem provided in the NCERT Solution of this chapter?

Yes. As these questions seem to be important for exams. These questions are solved by subject matter experts for helping students to crack these exercises easily. These solutions give students knowledge about data handling. Solutions can be downloaded in PDF format on the Infinity Learn website.

### List out the concepts covered in these NCERT Solutions?

The concepts included are 5.1 Looking for Information 5.2 Organising Data 5.3 Grouping Data 5.3.1 Bars with a difference 5.4 Circle Graph Or Pie Chart 5.4.1 Drawing pie charts 5.5 Chance and Probability 5.5.1 Getting a result 5.5.2 Equally likely outcomes 5.5.3 Linking chances to probability 5.5.4 Outcomes as events 5.5.5 Chances and Probability related to real life.

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