GSEB Class 8 Maths Solutions Chapter 3 Understanding Quadrilaterals Exercise 3.2

Question 1. Find x in the following figures.
(a)

(b)

Answer:
(a) The total sum of all the exterior angles of any polygon is always \( 360^\circ \). Therefore, \( 125^\circ + 125^\circ + x = 360^\circ \)
\( \implies 250^\circ + x = 360^\circ \)
\( \implies x = 360^\circ - 250^\circ \)
\( \implies x = 110^\circ \)
(b) The sum of all exterior angles of a polygon is \( 360^\circ \). Therefore, \( x + 90^\circ + 60^\circ + 90^\circ + 70^\circ = 360^\circ \)
\( \implies x + 310^\circ = 360^\circ \)
\( \implies x = 360^\circ - 310^\circ \)
\( \implies x = 50^\circ \)
In simple words: For any shape with straight sides, all the outside angles always add up to \( 360^\circ \). You just add up the angles you know and subtract from \( 360^\circ \) to find the missing one.

Exam Tip: Remember that the sum of the exterior angles of any convex polygon, regardless of the number of sides, is always \( 360^\circ \). This is a fundamental property to apply.

 

Question 2. Find the measure of each exterior angle of a regular polygon of
(i) 9 sides
(ii) 15 sides
Answer:
(i) Given, the number of sides (n) for the regular polygon is 9.
For a regular polygon, all its exterior angles are equal.
The total sum of all the exterior angles of any polygon is \( 360^\circ \).
So, the measure of each exterior angle \( = \frac{360^\circ}{\text{Number of sides}} \)
\( = \frac{360^\circ}{9} \)
\( = 40^\circ \)
(ii) Given, the number of sides for the regular polygon is 15.
For a regular polygon, all its exterior angles are equal in measure.
The total sum of all the exterior angles of any polygon is \( 360^\circ \).
So, the measure of each exterior angle \( = \frac{360^\circ}{\text{Number of sides}} \)
\( = \frac{360^\circ}{15} \)
\( = 24^\circ \)
In simple words: For a shape where all sides and angles are the same, you can find each outside angle by dividing \( 360^\circ \) by how many sides the shape has. This works for any regular polygon.

Exam Tip: Always remember that the sum of exterior angles is \( 360^\circ \). For a regular polygon, simply divide this sum by the number of sides to get each individual exterior angle.

 

Question 3. How many sides does a regular polygon have if the measure of an exterior angle is \( 24^\circ \)?
Answer: For any regular polygon, every exterior angle has the same measurement.
We know that the total sum of all exterior angles is \( 360^\circ \).
The measure of one exterior angle is given as \( 24^\circ \).
So, the number of sides \( = \frac{\text{Sum of all exterior angles}}{\text{Measure of one exterior angle}} \)
\( = \frac{360^\circ}{24^\circ} \)
\( = 15 \) Thus, this polygon has 15 sides.
In simple words: If you know what one outside angle of a regular shape is, you can find how many sides it has by dividing \( 360^\circ \) by that angle.

Exam Tip: This question is the inverse of Question 2. Ensure you understand that 'number of sides' is \( \frac{360^\circ}{\text{exterior angle}} \) and 'exterior angle' is \( \frac{360^\circ}{\text{number of sides}} \).

 

Question 4. How many sides does a regular polygon have if each of its interior angles is \( 165^\circ \)?
Answer: The polygon mentioned is a regular polygon.
Each interior angle measures \( 165^\circ \).
We know that an interior angle and its adjacent exterior angle always add up to \( 180^\circ \) (they form a linear pair).
So, each exterior angle \( = 180^\circ - \text{Interior Angle} \)
\( = 180^\circ - 165^\circ \)
\( = 15^\circ \) Now, using the rule from previous questions, the number of sides \( = \frac{\text{Sum of all exterior angles}}{\text{Measure of one exterior angle}} \)
\( = \frac{360^\circ}{15^\circ} \)
\( = 24 \) Therefore, this polygon has 24 sides.
In simple words: First, figure out the outside angle by taking the inside angle away from \( 180^\circ \). Then, divide \( 360^\circ \) by that outside angle to get the number of sides.

Exam Tip: Remember the relationship between interior and exterior angles (they sum to \( 180^\circ \)). This is key when the interior angle is provided instead of the exterior angle.

 

Question 5. (a) Is it possible to have a regular polygon with measure of each exterior angle \( 22^\circ \)?
(b) Can it be an interior angle of a regular polygon? Why?
Answer:
(a) If the measure of each exterior angle is \( 22^\circ \).
The number of sides of a regular polygon \( = \frac{360^\circ}{\text{Measure of each exterior angle}} \)
\( = \frac{360^\circ}{22^\circ} \)
\( = \frac{180}{11} \) For a polygon to be possible, its number of sides must be a whole number.
Since \( \frac{180}{11} \) is not a whole number (it's a fraction or decimal), it is not possible to have a regular polygon with an exterior angle of \( 22^\circ \).
(b) If \( 22^\circ \) were an interior angle of a regular polygon, then its corresponding exterior angle would be:
Exterior angle \( = 180^\circ - \text{Interior angle} \)
\( = 180^\circ - 22^\circ \)
\( = 158^\circ \) Now, let's find the number of sides with this exterior angle:
Number of sides \( = \frac{360^\circ}{\text{Measure of each exterior angle}} \)
\( = \frac{360^\circ}{158^\circ} \)
\( = \frac{180}{79} \) Again, \( \frac{180}{79} \) is not a whole number.
Therefore, \( 22^\circ \) cannot be an interior angle of a regular polygon either.
In simple words: A shape with straight sides must have a whole number of sides. If the calculations for either the outside or inside angle give you a fraction or decimal for the number of sides, then that angle isn't possible for a regular polygon.

Exam Tip: Always verify if the calculated number of sides is a whole number. This is a critical check for whether a polygon with specific angle measures can exist.

 

Question 6. (a) What is the minimum interior angle possible for a regular polygon? Why?
(b) What is the maximum exterior angle possible for a regular polygon?
Answer:
(a) The minimum number of sides a polygon can have is 3, which forms a triangle.
The simplest regular polygon is an equilateral triangle, which has 3 equal sides and 3 equal interior angles.
Each interior angle of an equilateral triangle is \( 60^\circ \).
If a polygon had more sides, its interior angles would become larger.
Hence, the minimum possible interior angle for a regular polygon is \( 60^\circ \).
(b) The sum of an exterior angle and its corresponding interior angle is always \( 180^\circ \).
To find the maximum exterior angle, we need the minimum interior angle.
From part (a), the minimum interior angle of a regular polygon is \( 60^\circ \).
So, the maximum exterior angle \( = 180^\circ - \text{Minimum interior angle} \)
\( = 180^\circ - 60^\circ \)
\( = 120^\circ \) Therefore, the maximum exterior angle possible for a regular polygon is \( 120^\circ \).
In simple words: The smallest number of sides a shape can have is three, making a triangle. This triangle has the smallest possible inside angles for any regular shape, which are \( 60^\circ \). The biggest outside angle will happen when the inside angle is the smallest, making it \( 180^\circ - 60^\circ = 120^\circ \).

Exam Tip: The minimum number of sides for a polygon is 3 (a triangle). This fact is crucial for determining minimum interior and maximum exterior angles for regular polygons.