RBSE Solutions Class 9 Maths Chapter 6 Rectilinear Figures Exercise 6.2

Chapter 6 Rectilinear Figures Ex 6.2

Question 1. A regular polygon has 8 sides, then
(i) Find the sum of the exterior angles.
(ii) Find the measure of each exterior angles.
(iii) Find the sum of all interior angles.
(iv) Find the measure of each interior angle.
Answer:
(i) The sum of all exterior angles for any polygon, including one with 8 sides, is always 360°. This is a basic rule for all convex polygons.
(ii) For an octagon (a polygon with 8 sides), each exterior angle is found by dividing the total sum of exterior angles by the number of sides:
\( = \frac { 360° }{ 8 } \)
\( = 45° \)
(iii) The sum of the interior angles of a polygon with 'n' sides is given by the formula \( (2n - 4) \times 90° \). For n = 8 sides, we calculate:
\( = (2 \times 8 - 4) \times 90° \)
\( = (16 - 4) \times 90° \)
\( = 12 \times 90° \)
\( = 1080° \)
(iv) Each interior angle of a regular octagon is found by dividing the sum of interior angles by the number of sides. We can also use the formula \( \frac { (2n - 4) }{ n } \times 90° \).
\( = \frac { (2 \times 8 - 4) }{ 8 } \times 90° \)
\( = \frac { (16 - 4) }{ 8 } \times 90° \)
\( = \frac { 12 }{ 8 } \times 90° \)
\( = \frac { 3 }{ 2 } \times 90° \)
\( = 3 \times 45° \)
\( = 135° \)
In simple words: For a shape with 8 equal sides: all its outside angles add up to 360 degrees, so each outside angle is 45 degrees. All its inside angles add up to 1080 degrees, so each inside angle is 135 degrees.

🎯 Exam Tip: Remember the two key formulas: sum of exterior angles is always 360°, and sum of interior angles is \( (n-2) \times 180° \) or \( (2n-4) \times 90° \). Always clearly state which formula you are using.

Question 2. The sum of the interior angles of a polygon is 2160°, find the number of sides in the polygon.
Answer: We know that the sum of the interior angles of a polygon with 'n' sides is given by the formula \( (2n - 4) \times 90° \). We are given that this sum is 2160°.
So, we can set up the equation:
\( (2n - 4) \times 90° = 2160° \)
First, divide both sides by 90°:
\( 2n - 4 = \frac { 2160 }{ 90 } \)
\( 2n - 4 = 24 \)
Now, add 4 to both sides:
\( 2n = 24 + 4 \)
\( 2n = 28 \)
Finally, divide by 2 to find 'n':
\( n = \frac { 28 }{ 2 } \)
\( n = 14 \)
Therefore, the polygon has 14 sides.
In simple words: We used a special math rule that connects the number of sides of a shape to the total degrees of its inside corners. By using this rule, we found that the shape must have 14 sides.

🎯 Exam Tip: When given the sum of interior angles, always use the formula \( (n-2) \times 180° \) or \( (2n-4) \times 90° \) to find 'n'. Ensure your calculations are careful, especially when dividing and rearranging the equation.

Question 3. Can there exist a regular polygon whose interior angle is 137°?
Answer: Let's assume, if possible, that there is a regular polygon with an interior angle of 137°. For any regular polygon, if its interior angle is 137°, then its exterior angle would be \( 180° - 137° = 43° \). The sum of all exterior angles of any polygon is always 360°. If 'n' is the number of sides of the polygon, then the measure of each exterior angle of a regular polygon is \( \frac { 360° }{ n } \).
So, we have:
\( 43° = \frac { 360° }{ n } \)
Now, we can find 'n' by rearranging the equation:
\( n = \frac { 360 }{ 43 } \)
\( n \approx 8.37 \)
Since the number of sides 'n' must be a whole number for a polygon to exist, a value of 8.37 sides is not possible. Thus, a regular polygon cannot have an interior angle of 137°. Every polygon must have an integer number of sides.
In simple words: No, a regular polygon cannot have an inside angle of 137 degrees. This is because if you calculate the number of sides it would need, it comes out as a fraction, and a shape must have a whole number of sides.

🎯 Exam Tip: Always check if 'n' (the number of sides) comes out as a whole number. If it's a fraction or a decimal, then such a polygon cannot exist. Using the exterior angle often simplifies the calculation.

Question 4. In the given figure, find the measure of ∠CED.
Answer: The problem asks to find ∠CED from the given figure. Looking at the figure and the solution provided, we will focus on the large triangle ACE.
The angle at vertex A (∠CAE) is 31°.
The angle at vertex C (∠ACE) is 75°.
We know that the sum of angles in any triangle is 180°. So, for triangle ACE:
\( \angle A + \angle C + \angle E = 180° \)
Substitute the known angle values:
\( 31° + 75° + \angle E = 180° \)
First, add the known angles:
\( 106° + \angle E = 180° \)
Now, subtract 106° from both sides to find ∠E:
\( \angle E = 180° - 106° \)
\( \angle E = 74° \)
Therefore, the measure of ∠CED (which is the same as ∠E or ∠AEC in this triangle) is 74°. Understanding the properties of triangles, like the sum of angles, helps solve many geometry problems.
In simple words: Inside any triangle, all three corners add up to 180 degrees. In triangle ACE, we know two corners (31 and 75 degrees). By taking these away from 180, we find the third corner, angle E, which is 74 degrees. This is also angle CED.

🎯 Exam Tip: When solving geometry problems with figures, always identify the relevant triangle or shape. The sum of angles in a triangle is a fundamental property often used to find unknown angles.