RBSE Solutions Class 8 Maths Chapter 6 Polygons Exercise 6.1

Question 1. Draw the diagonal with the help of a pencil in the following figure and tell
(i) In which shapes, diagonal will be inside?
(ii) In which shapes, diagonal will be outside?
(iii) Identify the type of polygon (Concave or Convex)?
Answer: The figures with their drawn diagonals are shown below for better understanding.

(i) Polygon with vertices E, A, B, C, D:
Diagonals: AC, AD, BD, BE, CE

(ii) Polygon with vertices A, B, C, D:
Diagonals: AC, BD

(iii) The figure is a convex quadrilateral. No separate redrawn image is provided in the solution, so it refers to the original figure (iii) from the question.

(iv) Polygon with vertices A, B, C, D:
Diagonals: AC, BD

(v) Polygon with vertices A, B, C, D, E, F, G, H:
Diagonals: AC, AD, AE, AF, AG, BD, BE, BF, BG, BH, CE, CF, CG, CH, DF, DG, DH, EG, EH, FH

(vi) Polygon with vertices A, B, C, D:

Diagonals: AC, AD, BD, BE, CE (Note: The provided list of diagonals contains items (BE, CE) that imply a pentagon, even though the drawn figure is a quadrilateral.)

Now, addressing the final sub-questions based on the original figures given in the question:
(i) Diagonals are completely inside the figures (iii), (iv), (v) and (vi). These are convex polygons.
(ii) Diagonals are outside the figures (i) and (ii). These are concave polygons.
(iii) Types of figures:
    (i) Concave
    (ii) Concave
    (iii) Convex
    (iv) Convex
    (v) Convex
    (vi) Convex
In simple words: For polygons where all diagonals are inside, they are convex. For polygons where some diagonals go outside, they are concave. The solution provides specific drawings for each part.

🎯 Exam Tip: Remember that a convex polygon has all its interior angles less than \(180^\circ\), and all its diagonals lie entirely within the polygon. A concave polygon has at least one interior angle greater than \(180^\circ\), and at least one diagonal will pass outside the polygon.

Question 2. Identify the interior and exterior angles shown in the figure.
Answer: In the given figure, which is a pentagon:
(i) The interior angles are labeled as \(a, b, c, d, e\). These angles are inside the polygon.
(ii) The exterior angles are labeled as \(p, q, r, s, t\). These angles are formed by extending one side of the polygon and the adjacent side.
In simple words: The angles that are inside the polygon are interior angles. The angles that are formed outside the polygon when you extend one side are exterior angles.

🎯 Exam Tip: For any polygon, an interior angle and its corresponding exterior angle at a vertex always add up to \(180^\circ\) (they form a linear pair).

Question 3. Define the regular polygon. Identify that regular polygon in which there are :
(i) 5 sides
(ii) 6 sides
(iii) 8 sides.
Answer: A regular polygon is a closed shape with three or more equal sides and equal angles. This means all sides have the same length, and all interior angles have the same measure.
(i) A closed figure with 5 equal sides and equal angles is called a pentagon.
(ii) A closed figure with 6 equal sides and equal angles is called a hexagon.
(iii) A closed figure with 8 equal sides and equal angles is called an octagon.
In simple words: A regular polygon is a shape where all its sides are the same length and all its angles are the same. A 5-sided one is a pentagon, a 6-sided one is a hexagon, and an 8-sided one is an octagon.

🎯 Exam Tip: Remember that "regular" means both equilateral (equal sides) and equiangular (equal angles). If only sides are equal, it's equilateral. If only angles are equal, it's equiangular. For regular polygons, both are true.

Question 4. Find the value of unknown angles (w, x, y, z) in the following figures.
Answer: We will find the unknown angles using the properties of polygons and angles.

(i) In a quadrilateral, the sum of all internal angles is \(360^\circ\).
The angles are \(x, 120^\circ, 130^\circ, 50^\circ\).
So, \(x + 120^\circ + 130^\circ + 50^\circ = 360^\circ\)
\( \implies x + 300^\circ = 360^\circ \)
\( \implies x = 360^\circ - 300^\circ \)
\( \implies x = 60^\circ \)

(ii) In a quadrilateral, the sum of all internal angles is \(360^\circ\).
The angles are \(x, 90^\circ, 60^\circ, 70^\circ\).
So, \(x + 90^\circ + 60^\circ + 70^\circ = 360^\circ\)
\( \implies x + 220^\circ = 360^\circ \)
\( \implies x = 360^\circ - 220^\circ \)
\( \implies x = 140^\circ \)

(iii) In a pentagon, the sum of all internal angles is given by the formula \( (n - 2) \times 180^\circ \). For \(n=5\) (pentagon), the sum is \( (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ \).
The given angles are \(30^\circ, x, (180^\circ - 70^\circ), (180^\circ - 60^\circ), x\).
So, \(30^\circ + x + (180^\circ - 70^\circ) + (180^\circ - 60^\circ) + x = 540^\circ\)
\( \implies 30^\circ + x + 110^\circ + 120^\circ + x = 540^\circ \)
\( \implies 2x + 260^\circ = 540^\circ \)
\( \implies 2x = 540^\circ - 260^\circ \)
\( \implies 2x = 280^\circ \)
\( \implies x = \frac{280}{2} \)
\( \implies x = 140^\circ \)

(iv) In a triangle, the sum of all angles is \(180^\circ\).
The angles are \(90^\circ, 30^\circ, (180^\circ - y)\).
So, \(90^\circ + 30^\circ + (180^\circ - y) = 180^\circ\)
\( \implies 120^\circ + 180^\circ - y = 180^\circ \)
\( \implies 300^\circ - y = 180^\circ \)
\( \implies y = 300^\circ - 180^\circ \)
\( \implies y = 120^\circ \)

Now, we find \(z\). The angle \(y\) and \(z\) are vertically opposite angles, so they are equal.
Thus, \(z = y = 120^\circ\).

(v) We have linear pairs of angles and the sum of internal angles of a quadrilateral.
For angle \(x\): \(x + 120^\circ = 180^\circ\) (linear pair)
\( \implies x = 180^\circ - 120^\circ \)
\( \implies x = 60^\circ \)
For angle \(y\): \(y + 80^\circ = 180^\circ\) (linear pair)
\( \implies y = 180^\circ - 80^\circ \)
\( \implies y = 100^\circ \)
For angle \(z\): \(z + 60^\circ = 180^\circ\) (linear pair)
\( \implies z = 180^\circ - 60^\circ \)
\( \implies z = 120^\circ \)
Now, for angle \(w\), the sum of all internal angles of a quadrilateral is \(360^\circ\).
The internal angles are \(180^\circ - w, 120^\circ, 80^\circ, 60^\circ\).
So, \( (180^\circ - w) + 120^\circ + 80^\circ + 60^\circ = 360^\circ \)
\( \implies 180^\circ - w + 260^\circ = 360^\circ \)
\( \implies 440^\circ - w = 360^\circ \)
\( \implies w = 440^\circ - 360^\circ \)
\( \implies w = 80^\circ \)
In simple words: We used two main rules: angles on a straight line add up to \(180^\circ\), and the total degrees inside shapes like triangles ( \(180^\circ\) ), quadrilaterals ( \(360^\circ\) ), and pentagons ( \(540^\circ\) ) are fixed. We added up all the known angles and subtracted them from the total to find the missing ones.

🎯 Exam Tip: Always identify the type of polygon first (triangle, quadrilateral, pentagon, etc.) to know the sum of its interior angles. Remember that an exterior angle and its adjacent interior angle form a linear pair and sum to \(180^\circ\).

Question 5. Find the number of sides of a regular polygon whose measure of each exterior angle is 45°.
Answer: Let \(n\) be the number of sides of the regular polygon. A key property of any regular polygon is that the sum of all its exterior angles is always \(360^\circ\).
Since it's a regular polygon, all its exterior angles are equal. Therefore, the measure of each exterior angle can be found by dividing the total sum by the number of sides.
So, Each exterior angle \( = \frac{360^\circ}{n} \).
We are given that each exterior angle is \(45^\circ\).
\( \implies 45^\circ = \frac{360^\circ}{n} \)
To find \(n\), we rearrange the equation:
\( \implies n = \frac{360^\circ}{45^\circ} \)
\( \implies n = 8 \)
So, the number of sides of the regular polygon is 8.
In simple words: All the outside angles of any regular shape add up to \(360^\circ\). Since each outside angle is \(45^\circ\), we divide \(360^\circ\) by \(45^\circ\) to find that the shape has 8 sides.

🎯 Exam Tip: Remember the formula for the measure of an exterior angle of a regular polygon: \( \frac{360^\circ}{n} \). This is a fundamental property for solving such problems quickly.

Question 6. Find the number of sides of a regular polygon if its each interior angle is 165°.
Answer: Let \(n\) be the number of sides of the regular polygon. We know that an interior angle and its corresponding exterior angle form a linear pair, meaning they add up to \(180^\circ\).
Given, each interior angle \( = 165^\circ \).
Therefore, each exterior angle \( = 180^\circ - \text{interior angle} \)
Each exterior angle \( = 180^\circ - 165^\circ = 15^\circ \).
The sum of all exterior angles of any polygon is \(360^\circ\).
For a regular polygon, each exterior angle \( = \frac{360^\circ}{n} \).
So, \( 15^\circ = \frac{360^\circ}{n} \)
\( \implies n = \frac{360^\circ}{15^\circ} \)
\( \implies n = 24 \)
Thus, the regular polygon has 24 sides.
In simple words: First, we find the outside angle by subtracting the inside angle from \(180^\circ\). Since all outside angles of any polygon add up to \(360^\circ\), we divide \(360^\circ\) by this outside angle to find the total number of sides.

🎯 Exam Tip: When given an interior angle, always find the exterior angle first. It simplifies calculations because the sum of exterior angles is a fixed \(360^\circ\) for any polygon, unlike interior angles which depend on the number of sides.

Question 7. Find the number of sides of that regular polygon whose each exterior angle is 24°.
Answer: Let \(n\) be the number of sides of the regular polygon. The sum of all exterior angles of any polygon is always \(360^\circ\).
For a regular polygon, all exterior angles are equal.
So, each exterior angle \( = \frac{360^\circ}{n} \).
We are given that each exterior angle is \(24^\circ\).
\( \implies 24^\circ = \frac{360^\circ}{n} \)
To find \(n\), we rearrange the equation:
\( \implies n = \frac{360^\circ}{24^\circ} \)
\( \implies n = 15 \)
Therefore, the number of sides of the regular polygon is 15.
In simple words: Since each outside angle of the regular shape is \(24^\circ\), and all outside angles together make \(360^\circ\), we divide \(360^\circ\) by \(24^\circ\) to find that the shape has 15 sides.

🎯 Exam Tip: This question is a direct application of the exterior angle formula. Ensure accurate division to get the correct number of sides.

Question 8. Find the value of every interior angle of that regular polygon which has 10 sides.
Answer: We are given that the number of sides of the regular polygon, \(n = 10\).
The value of each interior angle of a regular polygon can be found using the formula: \( \text{Each interior angle} = \frac{(n-2) \times 180^\circ}{n} \).
Substitute \(n = 10\) into the formula:
Each interior angle \( = \frac{(10-2) \times 180^\circ}{10} \)
\( = \frac{8 \times 180^\circ}{10} \)
\( = \frac{1440^\circ}{10} \)
\( = 144^\circ \)
Alternatively, we could first find the exterior angle: Each exterior angle \( = \frac{360^\circ}{n} = \frac{360^\circ}{10} = 36^\circ \). Then, each interior angle \( = 180^\circ - \text{exterior angle} = 180^\circ - 36^\circ = 144^\circ \). This method is often quicker.
Therefore, the value of each interior angle of a regular polygon with 10 sides is \(144^\circ\).
In simple words: To find the inside angle of a regular shape with 10 sides, we can use a formula that divides the total sum of inside angles by the number of sides. This shape has 10 equal inside angles, and each one is \(144^\circ\).

🎯 Exam Tip: Know both formulas: for the sum of interior angles \((n-2) \times 180^\circ\) and for each exterior angle \(\frac{360^\circ}{n}\). Often, calculating the exterior angle first and subtracting from \(180^\circ\) is simpler to find the interior angle.

Question 9. If interior angles of any polygon is 115° then will it be regular polygon?
Answer: For a polygon to be a regular polygon, all its interior angles must be equal, and the number of sides must be a whole number.
Given that each interior angle is \(115^\circ\).
First, find the measure of each exterior angle: Each exterior angle \( = 180^\circ - \text{interior angle} \)
Each exterior angle \( = 180^\circ - 115^\circ = 65^\circ \).
Now, we know that for any regular polygon, the number of sides \(n\) can be found using the formula: \( n = \frac{360^\circ}{\text{each exterior angle}} \).
So, \( n = \frac{360^\circ}{65^\circ} \)
\( n = \frac{72}{13} \)
\( n \approx 5.538 \)
Since the number of sides \(n\) is not a natural (whole) number, a polygon cannot have \( \frac{72}{13} \) sides. Therefore, it is not possible for a regular polygon to have an interior angle of \(115^\circ\). This means the given polygon is not a regular polygon, or such a regular polygon does not exist.
In simple words: If a shape had inside angles of \(115^\circ\), its outside angles would be \(65^\circ\). But when we calculate how many sides it would have, the answer is not a whole number. So, a regular shape cannot have an inside angle of \(115^\circ\).

🎯 Exam Tip: For a polygon to be "regular", the number of sides calculated using the exterior angle formula must always be a positive whole number. If it's a fraction or decimal, such a regular polygon does not exist.

Question 10. One interior angle of a hexagon is 165° and the measure of remaining interior angle is x° then find out the measure of all the angles.
Answer: A hexagon is a polygon with 6 sides, so \(n = 6\).
The sum of the interior angles of a hexagon is given by the formula: \( (n-2) \times 180^\circ \).
Sum of interior angles of a hexagon \( = (6-2) \times 180^\circ \)
\( = 4 \times 180^\circ \)
\( = 720^\circ \)
We are given that one interior angle is \(165^\circ\), and the remaining 5 interior angles each measure \(x^\circ\).
So, the sum of all interior angles can also be written as: \( 165^\circ + x^\circ + x^\circ + x^\circ + x^\circ + x^\circ \).
\( \implies 165^\circ + 5x^\circ = 720^\circ \)
To find \(x\), we subtract \(165^\circ\) from both sides:
\( \implies 5x^\circ = 720^\circ - 165^\circ \)
\( \implies 5x^\circ = 555^\circ \)
Now, divide by 5:
\( \implies x = \frac{555}{5} \)
\( \implies x = 111 \)
Therefore, the measure of the remaining interior angles is \(111^\circ\). The angles of the hexagon are \(165^\circ, 111^\circ, 111^\circ, 111^\circ, 111^\circ, 111^\circ\).
In simple words: First, we find the total degrees inside a 6-sided shape (hexagon), which is \(720^\circ\). We know one angle is \(165^\circ\), and the other five are all \(x\). So, we subtract \(165^\circ\) from \(720^\circ\) and then divide by 5 to find what \(x\) is.

🎯 Exam Tip: Always state the formula for the sum of interior angles of a polygon, \((n-2) \times 180^\circ\), and ensure correct substitution and arithmetic for full marks.

Question 11. By increasing the side of a triangle in a single direction, obtained exterior angles are 110°, 115° and x°, then find the value of x.
Answer: We know that for any convex polygon, the sum of its exterior angles is always \(360^\circ\). This property holds true for a triangle as well.
The three exterior angles of the triangle are given as \(110^\circ, 115^\circ, \text{and } x^\circ\).
So, \(110^\circ + 115^\circ + x^\circ = 360^\circ\)
\( \implies 225^\circ + x^\circ = 360^\circ \)
To find \(x\), subtract \(225^\circ\) from \(360^\circ\):
\( \implies x^\circ = 360^\circ - 225^\circ \)
\( \implies x^\circ = 135^\circ \)
Therefore, the value of \(x\) is \(135^\circ\).
In simple words: The three outside angles of any triangle always add up to \(360^\circ\). We are given two outside angles, so we add them up and subtract from \(360^\circ\) to find the missing third outside angle.

🎯 Exam Tip: A common mistake is to confuse interior and exterior angles. Always remember that the sum of exterior angles of *any* polygon (convex) is \(360^\circ\), regardless of the number of sides.

Question 12. Find the sum of all interior angles of a regular heptagon.
Answer: A heptagon is a polygon with 7 sides. So, the number of sides \(n = 7\).
The sum of all interior angles of any polygon is given by the formula: Sum \( = (n-2) \times 180^\circ \).
Substitute \(n = 7\) into the formula:
Sum of interior angles \( = (7-2) \times 180^\circ \)
\( = 5 \times 180^\circ \)
\( = 900^\circ \)
Therefore, the sum of all interior angles of a regular heptagon is \(900^\circ\). Each interior angle of a regular heptagon would be \( \frac{900^\circ}{7} \approx 128.57^\circ \).
In simple words: A heptagon has 7 sides. To find the total degrees inside it, we use a simple rule: subtract 2 from the number of sides, then multiply by \(180^\circ\). For a heptagon, this means \(5 \times 180^\circ\), which gives \(900^\circ\).

🎯 Exam Tip: Accurately identifying the number of sides for a given polygon name (e.g., heptagon = 7 sides) is crucial. Use the formula \((n-2) \times 180^\circ\) carefully to avoid calculation errors.