CBSE Class 8 Mathematics – 15. Quadrilaterals (Notes)

Understanding Quadrilaterals – Class 8 :

Quadrilateral: –  It is made by Latin words; Quardi + Latus.

Quadri – means four and Latus means side.

A polygon figure bounded by four sides is called quadrilateral.

A quadrilateral has 4 sides, 4 vertices and 4 angles.

The sum of the interior angles of a quadrilateral is equal to 360 degrees.

It is two dimensional (2-d) shapes.

The quadrilateral is basically of 6 types such as:

Angle sum property of a polygon:

Note- Sum of interior Angles of a polygon = (n – 2) × 180⁰     Where ‘n’ is the number of sides

Example:

1. Angle sum of a triangle = (3 – 2) × 180⁰ = 1 × 180⁰ = 180
2. Angle sum of a quadrilateral = (4 – 2) × 180⁰ = 2 × 180⁰ = 360⁰
3. Angle sum of a pentagon = (5 – 2) × 180⁰ = 3 × 180⁰ = 540⁰

Diagonals: The line segment which connects two non-consecutive vertices of a polygon is called diagonal.

For the figure shown above, SQ and PR are the two diagonals for the given quadrilateral.

Types of Quadrilaterals:

Adjacent sides – The sides of a quadrilateral with a common endpoint are called adjacent sides. In the given figure, (AB, BC), (BC, CD), (CD, DA) and (DA, AB) are four pairs of adjacent sides of quadrilateral ABCD.

Adjacent angles – Two angles of a quadrilateral having a common arm are called adjacent angles. Here, (∠D, ∠C), (∠C, ∠B), (∠B, ∠A), (∠A, ∠D) are adjacent angles

Opposite sides – Two sides of a quadrilateral which have no a common endpoint are called opposite sides. In the given figure, (AB, DC) and (AD, BC) are two pairs of opposite sides of quadrilateral ABCD.

Opposite angles -Two angles of a quadrilateral which have no common arm are called opposite angles. In the given figure, (∠A, ∠C) and (∠B, ∠D) are two pairs of opposite angles of quadrilateral ABCD.

Angle Sum Property of a Quadrilateral

According to the angle sum property of a Quadrilateral, the sum of all the four interior angles is 360 degrees.

 Proof: In the quadrilateral ABCD,

∠ABC, ∠BCD, ∠CDA, and ∠DAB are the internal angles.

AC is a diagonal

AC divides the quadrilateral into two triangles, ∆ABC and ∆ADC

We have learned that the sum of internal angles of a quadrilateral is 360°, that is, ∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°.

let’s prove that the sum of all the four angles of a quadrilateral is 360 degrees.

We know that the sum of angles in a triangle is 180°.

Now consider triangle ADC,

∠ADC + ∠DAC + ∠DCA = 180°                           (Sum of angles in a triangle)

Now consider triangle ABC,

∠ABC + ∠BAC + ∠BCA = 180°                            (Sum of angles in a triangle)

On adding both the equations obtained above we have,

(∠ADC + ∠DAC + ∠DCA) + (∠ABC + ∠BAC + ∠BCA) = 180° + 180°

∠ADC + (∠DAC + ∠BAC) + (∠BCA + ∠DCA) + ∠ABC = 360°

We see that (∠DAC + ∠BAC) = ∠DAB and (∠BCA + ∠DCA) = ∠BCD.

Replacing them we have,

∠ADC + ∠DAB + ∠BCD + ∠ABC = 360°

That is,

∠D + ∠A + ∠C + ∠B = 360°.

Or, the sum of angles of a quadrilateral is 360°. This is the angle sum property of quadrilaterals.