Understanding Quadrilaterals - 8th math

Understanding Quadrilaterals NCERT Exercise 3.2 Question (1) Find x in the following figures.

(a)

Solution

We know that, sum of exterior angles of a polygon = 360⁰

Thus, here sum of exterior angles of the given triangle

= 125⁰ + 125⁰ + x = 360⁰

⇒ 250⁰ + x = 360⁰

⇒ x = 360⁰ – 250⁰

⇒ x = 110⁰ Answer

(b)

Solution

Given, ∠ D = 90⁰

∠ F = 70⁰

∠ C = 60⁰

Thus, x = ?

Here, given, ∠ GAC = 90⁰

Now, since ∠ GAC and ∠ BAC forms a straight line, and hence are supplementary.

Thus, ∠ GAC + ∠ BAC = 180⁰

⇒ 90⁰ + ∠ BAC = 180⁰

⇒ ∠ BAC = 180⁰ – 90⁰

⇒ ∠ BAC = 90⁰

Now, we know that, sum of external angles of a polynomial = 360⁰

Thus, sum of angles of the given pentagon = 360⁰

⇒ 90⁰ + 70⁰ + x + 90⁰ + 60⁰ = 360⁰

⇒ 310⁰ + x = 360⁰

⇒ x = 360⁰ – 310⁰

⇒ x = 50⁰ Answer

Understanding Quadrilaterals NCERT Exercise 3.2 Question (2) Find the measure of each exterior angle of a regular polygon of

(i) 9 sides (ii) 15 sides

Solution

(i) Measure of each exterior angle of a regular polygon of 9 sides

We know that, each exterior angle of a regular polygon of n sides `360^o/n`

Where, n = number of sides

Thus, each exterior angle of a regular polygon of 9 sides `=360^o/9`

= 40⁰

Thus, each exterior angle of a regular polygon of 9 sides = 40⁰ Answer

(ii) Measure of each exterior angle of a regular polygon of 15 sides

We know that, each exterior angle of a regular polygon of n sides `360^o/n`

Where, n = number of sides

Thus, each exterior angle of a regular polygon of 15 sides `=360^o/15`

= 24⁰

Thus, each exterior angle of a regular polygon of 15 sides = 24⁰ Answer

Understanding Quadrilaterals NCERT Exercise 3.2 Question (3) How many sides does a regular polygon have if the measure of an exterior angle is 24⁰?

Solution

Given, measure of an exterior angle of a regular polygon = 24⁰

Therefore, number of sides of the given regular polygon = ?

We know that, number of sides of a regular polygon `360^o/x`

Where x = measure of each exterior angle of a regular polygon

Thus, number of sides of the given regular polygon `=360^o/24^o`

= 15 sides

Thus, number of sides of the given regular polygon = 15 Answer

Understanding Quadrilaterals NCERT Exercise 3.2 Question (4) How many sides does a regular polygon have if each of its interior angles is 165⁰?

Solution

Given, each of the interior angles of a regular polygon = 165⁰

Thus, number of sides = ?

Let each exterior angle = m

We know that, interior angle and exterior angle together form a straight line, and hence are supplementary.

Thus, interior angle + exterior angle = 180⁰

⇒ 165⁰ + m = 180⁰

⇒ m = 180⁰ – 165⁰

⇒ m = 15⁰

Thus, each exterior angle of the given polygon = 15⁰

Now, we know that, number of sides of a regular polygon `360^o/x`

Where x = measure of each exterior angle of a regular polygon

Thus, number of sides of the given regular polygon `=360^o/15^o`

= 24 sides

Thus, number of sides of the given regular polygon = 24 Answer

Understanding Quadrilaterals NCERT Exercise 3.2 Question (5) (a) Is it possible to have a regular polygon with measure of each exterior angle as 22⁰?

Solution

Given each exterior angle of a regular polygon = 22⁰

Thus, possibility of formation of regular polygon = ?

Now, we know that, number of sides of a regular polygon `360^o/x`

Where x = measure of each exterior angle of a regular polygon

Thus, number of sides of the given regular polygon `=360^o/22^o`

= 16.36 sides

Since, sum of exterior angles of a regular polygon which is equal to 360⁰ is not divisible completely by given angle of 22⁰, thus formation of such regular polygon is not possible.

Thus, Answer = No

(b) Can it be an interior angle of a regular polygon? Why?

Solution

If each interior angle of a regular polygon = 22⁰

Therefore, exterior angle of that very polygon = 180⁰ – 22⁰

= 158⁰

Here, 360⁰, which is the sum of exterior angles of a polygon is not divisible by 158⁰ completely.

Thus, it is not possible to have the measure of each exterior angle of a regular polygon to have equal to 158⁰.

Consequently, it is not possible to have each interior angle of a regular polygon equal to 22⁰.

Thus, Answer = No

Understanding Quadrilaterals NCERT Exercise 3.2 Question (6) (a) What is the minimum interior angle possible for a regular polygon? Why?

Solution

The measure of interior angle decreases with number of sides of a polygon.

We know that a triangle is a polygon with minimum number of sides.

A regular polygon with three sides is known as equilateral triangle.

And measure of each interior angle of a equilateral triangle is equal to 60⁰.

Thus, minimum interior angle possible for a regular polygon = 60⁰ Answer

(b) What is the maximum exterior angle possible for a regular polygon?

Solution

The maximum measure of exterior angle is possible when interior angle of a polygon is minimum.

Since, an equilateral triangle has minimum measure of interior angle and is equal to 60⁰

Thus, measure of exterior angle of an equilateral triangle

= 180⁰ – 60⁰

= 120⁰

Now, since an equilateral triangle has maximum measure of exterior angle and is equal to 120⁰

Thus, maximum measure of exterior angle possible for a regular polygon = 120⁰ Answer

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