A polygon is defined as any closed curve formed by straight lines connected in a way that no two lines cross.

Polygons are classified based on-

a) Number of sides

There are various types of polygons depending on the number of sides (lines), like triangles that consist of three sides, quadrilaterals of four sides, pentagons with five sides, polygons with six, and more.

b) Length of sides

Polygons are also classified as per the length of the sides, like the regular and irregular polygon.
In a regular polygon, all sides and interior angles are equal. For example, a square has all angles at 90 degrees and sides of equal length. In the same way, an equilateral triangle has angles at 60 degrees and equal sides.
Polygons in which the sides and interior angles are not equal are called irregular polygons such as rectangle and rhombus.

c) Degree of angle

A polygon is termed convex if all the interior angles are less than 180 degrees. The vertex points outwards from the centre of the shape
In a concave polygon, one or more of the interior angles is greater than 180 degrees. So here, concave polygons are the vertex points towards the inside of the polygon.

 

In order to attempt and understand quadrilaterals class 8 solutions, let us explore some of the polygons' significant concepts and properties;

The Angle sum Property

Interior Angles

The intersection of two sides forms an angle. Similarly, an angle is formed in a polygon (convex or non-convex) when two sides meet at a point (endpoint). This angle within the polygon is called the interior angle.
The sum of the interior angles can be calculated by the formula (n-2) x 180; n represents the number of sides in the polygon. It is assumed that the value of all the angles in a polygon is equal.

External Angles

When a transversal, i.e., a line passing through two lines at the same plane, intersects two parallel lines, external angles are formed.
In a polygon, an exterior angle is created by one of the sides meeting the extension of its closest (or the adjacent) line or side.

More Properties

• • The total sum of all the exterior angles of a polygon is 360 degrees.

  • The corresponding angles are equal.      

  • The vertically opposite angles are equal.

  • The alternate exterior angles are equal.

  • The interior angles on one side (or the same side) of the transversal are supplementary.

  • For a polygon, the sum of two angles in a linear pair is 180 degrees.

 

2.1 Introduction:

Having understood the basics of polygons, let us turn our attention to one more topic from the Class 8 Mathematics Chapter 3 – Quadrilaterals.

What is a quadrilateral? How is it connected to polygons? Simply put, a quadrilateral is a four-sided polygon.
Quadrilaterals come in various shapes like a square or a rectangle but always have four sides.

2.2 Types

The NCERT Maths Class 8 Chapter 3 specifies the study of the types of quadrilaterals.

1. Square: A square is a quadrilateral with all four sides equal in length, and the angles are at 90 degrees.

2. Kite: A kite is a four-sided quadrilateral where each pair of adjacent sides is equal in length. A square is a type of kite.

3. Rectangle: In a rectangle the opposite sides in a quadrilateral are equal, and the angles are at 90 degrees.

4. Parallelogram: In this case, the opposite sides are equal in length and parallel.

5. Rhombus: A unique parallelogram, which has all four sides as equal, and the diagonals bisect each other at 90 degrees.

6. Trapezium: In this type of quadrilateral, one pair of sides is parallel but not equal to each other.

2.3 Properties: Class 8 Mathematics understanding quadrilaterals

With the concept of a Polygon and Quadrilateral now clear, let us understand some of the assumptions or formulas for problem-solving:

In a quadrilateral, all the interior angles add up to a value of 360 degrees; similarly, the sum of all the exterior angles is also 360 degrees.

There are other critical properties of quadrilaterals to be remembered for your Class 8 Mathematics Chapter 3 studies:

• For example, the opposite sides of a parallelogram will always be equal.

• The two angles will be of equal measure always.

• The two diagonals bisect each other.

• The pair of adjacent angles in a parallelogram will always be supplementary angles.

• The properties of a rhombus and a parallelogram, and a square are similar.

• The diagonals of a rhombus are perpendicular bisectors (to each other).

• Rectangle.

• Opposite sides parallel and equal.

• Opposite angles are equal.

• Adjacent angles are a pair of supplementary angles.

• Diagonals are of equal length.

• Diagonals bisect each other.

• All four angles are 90 degrees.

The above properties from Class 8 Mathematics Chapter 3 lessons help answer various test questions.

Q1. How many vertices does a trapezium have?

There are four vertices in a trapezium.

Q2. What is a convex quadrilateral?

A quadrilateral is convex when the interior angles are less than 180 degrees; the vertex points outwards.

Q3. What is the degree –measure of each exterior angle of a polygon?

The value (degree) of each exterior angle of a polygon is 360 / n, which is the number of sides in the polygon (4, 5, 6, etc.). This is essential learning from the Class 8 Mathematics Chapter 3.

Q4. What is an equilateral triangle?

A triangle with all the sides of equal measure and all the angles of equal value (degree) is called an equilateral triangle.

 

Exercise 1

As per the figure given above, classify them based on-

                 I.          Simple curve

                II.          Simple closed curve

               III.          Polygon

               IV.          Convex Polygon

                V.          Concave Polygon

Solution:

                 I.          Simple curve: 1, 2, 5, 6 and 7

                II.          Simple closed curve: 1, 2, 5, 6 and 7

               III.          Polygon: 1 and 2

               IV.          Convex polygon: 2

                V.          Concave polygon: 1

Exercise 2

Calculate and tell the number of diagonals each of the following consists of?

    I.          Convex Quadrilateral

   II.          Regular Hexagon

  III.          Triangle

Solution:

I.          Convex Quadrilateral comprises of 2

 

 

  II.          Regular Hexagon has 9

 

 III.          Triangle has no diagonals, which makes it 0

Exercise 3

Calculate the sum of the measures of the angles of a convex quadrilateral? Will this property hold if the quadrilateral is not convex?

Solution

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 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, these properties hold if the quadrilateral is not convex.

The new-age syllabus of CBSE is designed to provide a clear concept of different topics and chapters. In addition, the e-learning app MSVgo facilitates interactive learning and helps students understand concepts easily. Whether it is Class 8 Mathematics Chapter 3 or any other, the innovative teaching methods help clarify all your doubts.