GSEB Class 8 Maths Solutions Chapter 3 ચતુષ્કોણની સમજ Exercise 3.2

Question 1. નીચેની આકૃતિઓમાં x શોધોઃ
(a)Answer: For this particular figure, the sum of all exterior angles is always \( 360^\circ \).
\( \implies x + 125^\circ + 125^\circ = 360^\circ \)
\( \implies x + 250^\circ = 360^\circ \)
\( \implies x = 360^\circ - 250^\circ \)
\( \implies x = 110^\circ \)
In simple words: When you add up all the outside angles of any shape, the total is always 360 degrees. Here, we know two outside angles are 125 degrees each, so we add them to 'x' and make the sum 360 degrees to find 'x'.

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

(b)Answer: In this specific figure, one of the exterior angles is \( 90^\circ \) and another is an interior angle of \( 90^\circ \). Therefore, there are two exterior angles that each measure \( 90^\circ \). The sum of all exterior angles in this figure is \( 360^\circ \).
\( \implies x + 90^\circ + 60^\circ + 90^\circ + 70^\circ = 360^\circ \)
\( \implies x + 310^\circ = 360^\circ \)
\( \implies x = 360^\circ - 310^\circ \)
\( \implies x = 50^\circ \)
In simple words: This shape has five outside angles. We know four of them. Because all outside angles must add up to 360 degrees, we can add the known ones and subtract from 360 to find the missing 'x' angle.

Exam Tip: Be careful with figures that show both interior and exterior angles. Always convert interior angles to their corresponding exterior angles ( \( 180^\circ \) - interior angle) before summing them up if the question asks for exterior angles.

Question 2. નીચે પ્રમાણેની બાજુઓ ધરાવતા નિયમિત બહુકોણમાં બહિષ્કોણનું માપ શોધોઃ
(a) 9 બાજુ
Answer: For a polygon with 9 sides:
Here, the polygon has 9 sides. So, we will consider \( n = 9 \).
A 9-sided polygon has 9 exterior angles.
This is a regular polygon, so all its exterior angles have the same measure.
Now, the sum of all exterior angles of a polygon is \( 360^\circ \).
\( \implies \) Therefore, the measure of each exterior angle of a 9-sided polygon \( = \frac{360^{\circ}}{n} = \frac{360^{\circ}}{9} = 40^\circ \)
In simple words: If a polygon has 9 equal sides, it means it has 9 equal outside angles. Since all outside angles always add up to 360 degrees, we divide 360 by 9 to find out how big each angle is.

Exam Tip: For any regular polygon, the measure of each exterior angle can be found by dividing \( 360^\circ \) by the number of sides (n). This formula is important for calculations.

(b) 15 બાજુ
Answer: For a polygon with 15 sides:
Here, the polygon has 15 sides. So, we will consider \( n = 15 \).
A 15-sided polygon has 15 exterior angles.
This is a regular polygon, so all its exterior angles have the same measure.
Now, the sum of all exterior angles of a polygon is \( 360^\circ \).
\( \implies \) Therefore, the measure of each exterior angle of a 15-sided polygon \( = \frac{360^{\circ}}{n} = \frac{360^{\circ}}{15} = 24^\circ \)
In simple words: If a regular polygon has 15 sides, it also has 15 equal outside angles. Since all outside angles always total 360 degrees, we divide 360 by 15 to find the size of each angle.

Exam Tip: Always state the formula ( \( \frac{360^{\circ}}{n} \) ) and show the substitution of 'n' before giving the final answer. This demonstrates your understanding of the process.

Question 3. એક નિયમિત બહુકોણને કેટલી બાજુઓ હોય તો તેના દરેક બહિષ્કોણનું માપ 24° થાય?
Answer: Here, the polygon is regular, so all its exterior angles have the same measure.
The sum of all exterior angles of a polygon is \( 360^\circ \).
Now, the measure of the exterior angle of this polygon is \( 24^\circ \).
\( \implies \) Therefore, the number of sides of this polygon \( = \frac{360^{\circ}}{24^{\circ}} = 15 \).
A regular polygon has as many sides as it has angles.
\( \implies \) This polygon has a total of 15 sides.
In simple words: Since all outside angles of any polygon add up to 360 degrees, and each outside angle of this specific regular polygon is 24 degrees, we can divide 360 by 24 to find how many sides the polygon has.

Exam Tip: When given the measure of an exterior angle of a regular polygon and asked for the number of sides, remember that \( n = \frac{360^\circ}{\text{exterior angle}} \).

Question 4. એક નિયમિત બહુકોણને કેટલી બાજુઓ હોય તો તેના દરેક અંતઃકોણનું માપ 165° થાય?
Answer: The polygon is regular, and each of its interior angles measures \( 165^\circ \).
\( \implies \) The measure of each exterior angle of the polygon \( = 180^\circ - 165^\circ = 15^\circ \).
The sum of all exterior angles of a polygon is \( 360^\circ \).
\( \implies \) Therefore, the number of sides of the polygon \( = \frac{360^{\circ}}{15^{\circ}} = 24 \).
\( \implies \) This polygon has a total of 24 sides.
In simple words: First, we find the outside angle by subtracting the inside angle from 180 degrees. Then, because all outside angles always sum to 360 degrees, we divide 360 by that outside angle to find the number of sides.

Exam Tip: Always convert the given interior angle to an exterior angle first, as the formula for the number of sides directly uses the exterior angle. The relationship is: interior angle + exterior angle = \( 180^\circ \).

Question 5. (a) એવો નિયમિત બહુકોણ શક્ય છે કે જેમાં દરેક બહિષ્કોણનું માપ 22° હોય?
Answer: The measure of the exterior angle of this polygon is \( 22^\circ \).
\( \implies \) Therefore, the number of sides of the polygon \( = \frac{360^{\circ}}{22^{\circ}} = \frac{180}{11} \).
If this polygon were a regular polygon, its number of sides would need to be a whole number.
Observe that \( \frac{180}{11} \) is not a whole number.
\( \implies \) No, it is not possible for a regular polygon to have an exterior angle of \( 22^\circ \).
In simple words: To find the number of sides of a polygon from its outside angle, you divide 360 by that angle. If the answer isn't a whole number, then such a polygon cannot exist because a polygon must have a whole number of sides.

Exam Tip: For a regular polygon, the number of sides must always be a positive integer. If calculations result in a fractional number of sides, such a polygon is not geometrically possible.

(b) શું આ માપ નિયમિત બહુકોણના અંતઃકોણનું હોઈ શકે? કેમ?
Answer: The measure of the interior angle of this polygon is \( 22^\circ \).
\( \implies \) The measure of the exterior angle \( = 180^\circ - 22^\circ = 158^\circ \).
\( \implies \) The number of sides of the polygon \( = \frac{360^{\circ}}{158^{\circ}} = \frac{180}{79} \).
If this polygon were a regular polygon, its number of sides would need to be a whole number.
Observe that \( \frac{180}{79} \) is not a whole number.
\( \implies \) No, it is not possible for a regular polygon to have an interior angle of \( 22^\circ \).
In simple words: If the inside angle is 22 degrees, the outside angle would be 158 degrees. When you try to find the number of sides by dividing 360 by 158, you get a fraction, not a whole number. Since a polygon must have a whole number of sides, an inside angle of 22 degrees is not possible for a regular polygon.

Exam Tip: Always remember to check if the calculated number of sides is a whole number. If it's not, then the given angle (whether interior or exterior) cannot belong to a regular polygon.

Question 6. (a) નિયમિત બહુકોણમાં અંતઃકોણનું ઓછામાં ઓછું માપ કેટલું હોઈ શકે? કેમ?
Answer: The minimum number of sides for a regular polygon is 3.
\( \implies \) A regular polygon with 3 sides is an equilateral triangle.
Each angle of an equilateral triangle measures \( 60^\circ \).
\( \implies \) Therefore, the minimum measure of each interior angle of a regular polygon can be \( 60^\circ \).
In simple words: The smallest number of sides a regular polygon can have is three, which forms an equilateral triangle. All angles in an equilateral triangle are 60 degrees. So, 60 degrees is the smallest possible inside angle for any regular polygon.

Exam Tip: An equilateral triangle is the regular polygon with the fewest sides, and thus its interior angle of \( 60^\circ \) represents the minimum possible interior angle for any regular polygon.

(b) નિયમિત બહુકોણમાં બહિષ્કોણનું વધુમાં વધુ માપ કેટલું હોઈ શકે?
Answer: The sum of each interior angle and its corresponding exterior angle in a regular polygon is \( 180^\circ \).
Now, the minimum measure of each interior angle of a regular polygon is \( 60^\circ \).
\( \implies \) Therefore, the maximum measure of the exterior angle of the polygon can be \( 180^\circ - 60^\circ = 120^\circ \).
In simple words: The biggest outside angle a regular polygon can have happens when its inside angle is the smallest. Since the smallest inside angle is 60 degrees, the biggest outside angle will be 180 minus 60 degrees.

Exam Tip: The exterior angle is maximized when the interior angle is minimized. Since the minimum interior angle for a regular polygon is \( 60^\circ \) (from an equilateral triangle), the maximum exterior angle will be \( 180^\circ - 60^\circ = 120^\circ \).