NCERT Solutions for Class 8 Maths Chapter 3: Understanding Quadrilaterals - Exercise 3.1

Exercise 3.1 

1. Given here are some figures, 

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Classify the following figures on the basis of the following 

1. Simple curve 

Ans: From the given figures, the figures with simple curves are figure1,2,5,6,7. 

2. Simple closed curve 

Ans: From the given figures, the figures with simple closed curves are figure  1,2,5,6,7. 

3. Polygon 

Ans: From the given figures, the figures that are polygons are figures 1,2 .

4. Convex polygon 

Ans: From the given figures, the figures that are convex polygon is figure 2 . 

5. Concave polygon 

Ans: From the given figures, the figures that are concave polygon is figure 1. 

2. How man diagonal does each of the following have? 

1. A convex quadrilateral 

Ans: Let’s first draw a convex quadrilateral, 

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As we can see that only 2diagonals can be made, 

Therefore, a convex quadrilateral has two diagonals. 

2. A regular hexagon 

Ans: Let’s first draw a regular hexagon, 

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As we can see that 9diagonals can be made, 

Therefore, a regular hexagon has nine diagonals. 

3. A triangle 

Ans: Let’s first draw a triangle,

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As we can see that no diagonals can be made, 

Therefore, a triangle has no or zero diagonals. 

3. What is the sum of the measure of the angles of a convex quadrilateral?  Will this property hold if the quadrilateral is not convex? (Make a non convex quadrilateral and try) 

Ans: First let us draw a convex quadrilateral, 

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Now, we know that a convex quadrilateral is made up of two triangles, also, the  sum of angles of a triangle is 180𝆩. 

Now, we know ABCDis a convex quadrilateral which is made up of two  triangles (ΔABD and ΔBCD), so, the sum of the measures of the angles of a  convex quadrilateral will be 360𝆩.

. 

Also, this property will hold even when the quadrilateral is not convex as we  know that any quadrilateral can be divided into two triangles. 

4. 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 the convex polygon with number  of sides?

(a 7) 

(b 8) 

(c 10 ) 

(d n) 

Ans: Now, examining the table carefully, we can see there’s a connection  between them, that is, the angle sum of a convex polygon of nside is given as(n-2) $\times$  180𝆩, (n is number of sides) now we can find the angle sum of the  convex polygon of the following 

1. 7

Ans: = (7-2) $\times$  180𝆩 = 5 $\times $   180𝆩

= 900𝆩

2. 8 

Ans: (8-2) $\times$  180𝆩 = 6 $\times $   180𝆩

=1080𝆩

3. 10
   

Ans: (n-2) $\times$  180𝆩 = 8 $\times $   180𝆩

=1440𝆩

4. n

Ans: (n-2) $\times$  180𝆩

5. What is a regular polygon?  

State and name of a regular polygon of 

(i )3sides 

(ii )4sides 

(iii )6sides 

Ans: When a polygon has equal sides and equal angles, then we can say that the  polygon is a regular polygon. 

(i ) 3sides

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Ans: We know that a polygon with three equal sides and angles is an equilateral  triangle 

(ii) 4sides 

Ans: We know that a polygon with four equal sides and angles is a square 

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(iii) 6 sides 

Ans: We know that a polygon with sic equal sides and equal angles is a regular  hexagon 

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6. Find the angle measure xin the following figures 

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Ans: Now, we know that the sum of the interior angles of a quadrilateral is 360°

Now, from the given figure we got that 50° +130° +120° +x=360°

⇒300° + x = 360°

X = 60°

We got x = 60°

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Ans: Now, in the figure we can say that 90° +a=180°(Linear pair) so, 

a=90°

Now, as we know that the sum of the interior angles of a quadrilateral is 

360° ,  

we got 

60° +70° +90° +x=360° 

⇒220°  + x = 360° 

= x= 140°

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Ans: Now, in the figure we can say that 

70° +a=180°(Linear pair) and 60° +b=180°(Linear pair), we got a=110° and b= 120°

Now, as we know that the sum of the interior angles of a pentagon is 540°, we  

got 

120° +110° +30° +x+x=540 

⇒ 220° +2x=360

=2x = 280°

=x = 140°

We got x=140° 

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Ans: Now, as we can see in the figure, it is a regular pentagon, and we know  that the sum of the interior angles of a pentagon is 540°, we got 

x+x+x+x+x=540°

⇒5x =540°

=x=108° 

We got x=108° 

7. Find 

1. x+y+z  

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Ans: From the given figure we got, 

⇒ x+90° =180°

            (As it’s linear pair) 

=x=90°

⇒z+30° =180°

      (As it’s linear pair)

= z = 150°

And

⇒y+ 60° = 180°

=y=120°(From exterior angle theorem) 

Now we got the required values, so, 

x+y+z=90°+120°+150° 

⇒ x+y+z=360° 

(b) w+x+y+z  

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Ans: 

Now, as we know that the sum of the interior angles of a quadrilateral is 360° ,  

we got 

60° +80° +120° +a=360° 

⇒ 260°  + a =360° 

= a = 100° 

Now, from the given figure we got

⇒x+120°  =180°  (As its Linear Pair)

=x=60°  

⇒ y+80°  =180°    (As Its Linear Pair)

= y = 100°

⇒ z+60° =180°   (As it’s linear pair)

= z = 120°

⇒ w+100° =180°  (As it’s linear pair)

= w = 80°

Now we got the required values, so, w+x+y+z=80° +60° +100° +120° 

⇒ w+x+y+z=360°

What is a Plane Curve? 

When you draw a few dots on a plane surface and connect them without raising your hand or retracing your steps, the resulting figure is called a plane curve. You have already studied the different types of plane curves(Simple closed, closed curve, Simple curve, etc.) in earlier classes. 

What is a Polygon? 

By definition, a polygon is a simple closed curve made up of only line segments. Some common examples of polygons are square, rectangle, square, etc. 

Classifications of Polygons

Based on the number of sides in a polygon, it can be classified into different shapes. A three-sided polygon is called a triangle, four-sided is called quadrilateral, and so forth. Read the below definition, they will come handy in ex 3.1 Class 8 NCERT Solutions.

Diagonals - A line segment that joins two non-consecutive (occurring one after the other) vertices in a polygon is called a diagonal. 

Interior & Exterior - The area enclosed in the boundary of the polygon is called interior while the area outside is called exterior. 

Concave Polygon and Convex Polygon - All the diagonals lie in the interior of a concave polygon and no two line segments join the same vertices, whereas in convex polygon a diagonal can lie in both the interior and exterior portion of the polygon and more than one line segments can meet at one vertex. 

Irregular & Regular Polygons 

A polygon is a regular polygon when it has the following features: 

• The length of all sides is equal.

• All angles are equal

A square and equilateral triangle both are regular polygons. On the other hand, a rectangle is not, because all the angles are equal to one another(90 °) but the length of all sides is not the same. Such a polygon is called an irregular polygon. If you understand the difference, you wouldn’t need exercise 3.1 Class 8 NCERT Solutions for comparison questions. 

Angle Sum Property 

The angle sum property of a triangle states that the sum of all the three angles is equal to 180°. You will apply it in a number of questions in Exercise 3.1.

NCERT Solutions for Class 8 Maths Chapter 3 Exercise 3.1

If you are looking for NCERT Solutions for Class 8 Maths Chapter 3 exercise 3.1, the below hints will certainly help you. 

Question 1. 

In the first question you are asked to classify the figure into the given options. 

1. Simple Curve -It is a type of curve that doesn’t cross itself. Figure(1) is a simple curve because it doesn’t close itself. Hence, it is a simple curve. In this way you can compare the rest of the figures and find which figure can be classified under which option. If you need NCERT Math book Class 8 exercise 3.1 solution along with complete explanations, Vedantu has it all. 

Question 2. 

You are asked the number of diagonals the quadrilaterals have. The first option is (a)Convex Quadrilateral - Since, diagonals lie in the interior of a convex quadrilateral, and no line segment can join two different points, it will have two diagonals. Similarly, you can solve the rest of the parts of the question of Class 8 Maths Chapter 3 exercise 3.1. If you are finding it difficult to figure out the solution of NCERT Maths Cass 8 exercise 3.1, you can contact our experts. 

Question 3.

The question asks you to define the sum of angles in a convex quadrilateral. Let ABCD be a convex quadrilateral and AC be diagonal. As you can see AC divides the quadrilateral into two triangles. According to the angle sum property, the sum of all angles in a triangle is 180. Thus, the sum of all the angle in △ ABC =△ ADC = 180°. After adding all the angles formed by both the triangle you will find the measure of a convex quadrilateral. 

For non-convex quadrilateral, checkout Vedantu ex 3.1 Class 8 free pdf. It has covered the solution to our questions in the exercise. 

Question 4. 

As you can see from the table, the sum of the angles in a polygon with n sides = (n-2 )X 180 °. Applying this formula you can easily solve all the option in the question. For your convenience, we have explained part (a) below. 

For (a) n = 7 

Therefore, the sum of the angles will be (7-2) x 180 ° = 5 X 180 ° = 900 °. Still in doubt? Review our NCERT Maths Class 8 Chapter 3 exercise 3.1 solutions. The step-by-step solution along with the simple explanations make learning math a joyride. 

Question 5. 

Regular polygon - You are already familiar with it. It is a polygon which is both equilateral and equiangular.  

(i) Think about a polygon having three sides of equal length and equal angle. 

Find it hard to solve NCERT Maths Class 8 Chapter 3 exercise 3.1? Read the solutions from our experts. 

Question 6. 

The addition of all the interior angles in a quadrilateral is 360°. So, to find the value of ‘x’ in the question, you have to subtract the sum of all the remaining angles from 360° to find the value of x. 

1. Here 50°+360°+120°+x = 180° Solving this equation further, you will find the solution to this part. Use the same approach to solve (b),(c), and (d). Find it hard? Get a detailed explanation of exercise 3.1 Class 8 NCERT Solutions from our tutors. 

Question 7.  

Here we have to use three concepts, the angle sum property, exterior angle theorem and linear pair. 

Angle Sum Property -We have already discussed the angle sum property a couple of times above.

Exterior Angle Theorem - The measure of an exterior angle is the sum of the adjacent angle and an interior angle opposite to it. 

Linear Pair - When two adjacent angles make a single line, their sum = 180°, and they are called linear pair. 

Download NCERT Maths Class 8 to help you to revise complete syllabus ans score more marks in your examinations.

Understanding quadrilaterals exercise 3.1 at Vedantu

Every student wants to brush up his mathematics and score top grades in his exams. At times, it can become a challenge to find the right answers. The students are introduced to the various aspects of a polygon(diagonal, interior, and exterior). Our NCERT Solutions for Class 8 Maths Chapter 3 is written by the subject matter experts. Most pupils who are weak in math lag behind because they still haven’t gotten the grasp of basics. The ex 3.1 Class 8 ‘Understanding quadrilaterals’ tests student’s knowledge of polygons and their classification. We provide free chapter-wise solution and study material to help students in their self-study. If you are more comfortable with video lesson, you can view Understanding quadrilaterals exercise 3.1. Everything you read in the understanding quadrilaterals Class 8 exercise 3.1 study materials are chosen and put together by our experts. For improving student's academic performance we have created study courses that they can pace as per their preference.