Understanding Shapes

Q1: Multiple Choice Type:

i. Each interior angle of a regular polygon is double of its exterior angle, the number of sides in the polygon is:

Step 1: Let each exterior angle be \( x^\circ \). Then each interior angle is \( 2x^\circ \).
Step 2: Since interior and exterior angles are supplementary for a polygon, \[ x + 2x = 180^\circ \Rightarrow 3x = 180^\circ \Rightarrow x = 60^\circ \]Step 3: Number of sides \( n = \frac{360^\circ}{\text{exterior angle}} = \frac{360^\circ}{60^\circ} = 6 \).
Answer: c. 6

ii. The sum of all interior angles of a regular polygon is 1260°; then the number of sides in the polygon is:

Step 1: Sum of interior angles formula: \[ (n – 2) \times 180^\circ = 1260^\circ \]Step 2: Solve for \( n \): \[ n – 2 = \frac{1260^\circ}{180^\circ} = 7 \\ n = 7 + 2 = 9 \]Answer: c. 9

iii. The sum of all interior angles of a regular polygon is equal to sum of all its exterior angles. The number of sides in the polygon is:

Step 1: Sum of exterior angles of any polygon is \( 360^\circ \). Let sum of interior angles also be \( 360^\circ \).
Step 2: Use the formula: \[ (n – 2) \times 180^\circ = 360^\circ \]Step 3: Solve for \( n \): \[ n – 2 = \frac{360^\circ}{180^\circ} = 2 \\ n = 2 + 2 = 4 \]Answer: a. 4

iv. If each interior angle of a regular polygon is \(1\frac{1}{3}\) right angle. The number of sides in the polygon is:

Step 1: Convert the interior angle from right angles to degrees.
One right angle = 90°
Interior angle = 1 1/3 × 90°
Interior angle = (4/3) × 90°
Interior angle = 4 × 30° = 120°
Step 2: Find the measure of each exterior angle.
Exterior angle = 180° – Interior angle
Exterior angle = 180° – 120° = 60°
Step 3: Calculate the number of sides (n).
Formula: Number of sides = 360° / Exterior angle
n = 360° / 60°
n = 6
Answer: c. 6

v. The ratio between each interior angle of a regular polygon and each exterior angle of it is 3 : 2. The number of sides in the polygon is:

Step 1: Let the interior angle be \( 3k \) and exterior angle be \( 2k \). Since interior and exterior angles are supplementary: \[ 3k + 2k = 180^\circ \Rightarrow 5k = 180^\circ \Rightarrow k = 36^\circ \]Step 2: Exterior angle \( = 2k = 2 \times 36^\circ = 72^\circ \). Number of sides: \[ n = \frac{360^\circ}{72^\circ} = 5 \]Step 3: Check options: 5 is not listed, so answer is “none of these”.
Answer: d. none of these (5 sides)

Q2: Fill in the blanks:

Formula Used:
For a regular polygon of n sides:
Each exterior angle = \[ \frac{360^\circ}{n} \]Each interior angle = \[ 180^\circ – \text{exterior angle} \]

i. Number of sides = 8

Exterior angle: \[ = \frac{360^\circ}{8} = 45^\circ \] Interior angle: \[ = 180^\circ – 45^\circ = 135^\circ \]Answers: 8 sides → Exterior = 45°, Interior = 135°

ii. Number of sides = 12

Exterior angle: \[ = \frac{360^\circ}{12} = 30^\circ \] Interior angle: \[ = 180^\circ – 30^\circ = 150^\circ \]Answers: 12 sides → Exterior = 30°, Interior = 150°

iii. Each exterior angle = 72°

Number of sides: \[ = \frac{360^\circ}{72^\circ} = 5 \] Interior angle: \[ = 180^\circ – 72^\circ = 108^\circ \]Answers: 5 sides → Exterior = 72°, Interior = 108°

iv. Each exterior angle = 45°

Number of sides: \[ = \frac{360^\circ}{45^\circ} = 8 \] Interior angle: \[ = 180^\circ – 45^\circ = 135^\circ \]Answers: 8 sides → Exterior = 45°, Interior = 135°

v. Each interior angle = 150°

Exterior angle: \[ = 180^\circ – 150^\circ = 30^\circ \] Number of sides: \[ = \frac{360^\circ}{30^\circ} = 12 \]Answers: 12 sides → Exterior = 30°, Interior = 150°

vi. Each interior angle = 140°

Exterior angle: \[ = 180^\circ – 140^\circ = 40^\circ \] Number of sides: \[ = \frac{360^\circ}{40^\circ} = 9 \]Answers: 9 sides → Exterior = 40°, Interior = 140°

Q3: Find the number of sides in a regular polygon, if each interior angle is:

i. 160°

Step 1: Exterior angle \( = 180^\circ – \) interior angle = \(180^\circ – 160^\circ = 20^\circ\).
Step 2: Number of sides \( n = \frac{360^\circ}{\text{exterior angle}} = \frac{360^\circ}{20^\circ} = 18 \).
Answer: The polygon has 18 sides.

ii. 135°

Step 1: Exterior angle \( = 180^\circ – 135^\circ = 45^\circ\).
Step 2: Number of sides \( n = \frac{360^\circ}{45^\circ} = 8 \).
Answer: The polygon has 8 sides.

iii. \(1\frac{1}{5}\) of a right angle

Step 1: Convert \(1\frac{1}{5}\) to improper fraction: \[ 1\frac{1}{5} = \frac{6}{5} \] Right angle \(= 90^\circ\), so interior angle \(= \frac{6}{5} \times 90^\circ = 108^\circ\).
Step 2: Exterior angle \(= 180^\circ – 108^\circ = 72^\circ\).
Step 3: Number of sides \( n = \frac{360^\circ}{72^\circ} = 5 \).
Answer: The polygon has 5 sides.

Q4: Find the number of sides in a regular polygon, if each exterior angle is:

i. \(\frac{1}{3}\) of a right angle

Step 1: Right angle \(= 90^\circ\), so exterior angle \(= \frac{1}{3} \times 90^\circ = 30^\circ\).
Step 2: Number of sides \( n = \frac{360^\circ}{\text{exterior angle}} = \frac{360^\circ}{30^\circ} = 12 \).
Answer: The polygon has 12 sides.

ii. Two-fifths of a right angle

Step 1: Exterior angle \(= \frac{2}{5} \times 90^\circ = 36^\circ\).
Step 2: Number of sides \( n = \frac{360^\circ}{36^\circ} = 10 \).
Answer: The polygon has 10 sides.

Q5: Is it possible to have a regular polygon whose each interior angle is:

i. 170°

Step 1: Calculate the exterior angle: \[ \text{Exterior angle} = 180^\circ – 170^\circ = 10^\circ \]Step 2: Number of sides \( n = \frac{360^\circ}{10^\circ} = 36 \).
Step 3: Since \( n \) is a whole number greater than 2, such a polygon is possible.
Answer: Yes, a regular polygon with each interior angle 170° is possible (with 36 sides).

ii. 138°

Step 1: Calculate the exterior angle: \[ \text{Exterior angle} = 180^\circ – 138^\circ = 42^\circ \]Step 2: Number of sides \( n = \frac{360^\circ}{42^\circ} \approx 8.57 \).
Step 3: Since \( n \) is not a whole number, such a regular polygon is not possible.
Answer: No, a regular polygon with each interior angle 138° is not possible.

Q6: Is it possible to have a regular polygon whose each exterior angle is:

i. 80°

Step 1: Calculate the number of sides \( n \): \[ n = \frac{360^\circ}{80^\circ} = 4.5 \]Step 2: Since \( n \) is not a whole number, a regular polygon with each exterior angle 80° is not possible.
Answer: No, such a regular polygon is not possible.

ii. 40% of a right angle

Step 1: Right angle \(= 90^\circ\), so exterior angle \(= 0.4 \times 90^\circ = 36^\circ\).
Step 2: Calculate the number of sides \( n \): \[ n = \frac{360^\circ}{36^\circ} = 10 \]Step 3: Since \( n \) is a whole number greater than 2, such a polygon is possible.
Answer: Yes, a regular polygon with each exterior angle 36° (40% of a right angle) is possible with 10 sides.

Q7: Find the number of sides in a regular polygon, if its interior angle is equal to its exterior angle.

Step 1: Let the interior angle = exterior angle = \( x^\circ \).
Step 2: Since interior and exterior angles are supplementary (sum to 180°), \[ x + x = 180^\circ \Rightarrow 2x = 180^\circ \Rightarrow x = 90^\circ \]Step 3: Each exterior angle is \( 90^\circ \). Number of sides \( n \) is: \[ n = \frac{360^\circ}{90^\circ} = 4 \]Answer: The polygon has 4 sides.

Q8: The exterior angle of a regular polygon is one-third of its interior angle. Find the number of sides in the polygon.

Step 1: Let the interior angle be \( x^\circ \). Then the exterior angle is \(\frac{x}{3}^\circ\).
Step 2: Since interior and exterior angles are supplementary: \[ x + \frac{x}{3} = 180^\circ \]Step 3: Multiply both sides by 3 to clear denominator: \[ 3x + x = 540^\circ \Rightarrow 4x = 540^\circ \]Step 4: Solve for \( x \): \[ x = \frac{540^\circ}{4} = 135^\circ \]Step 5: Find the exterior angle: \[ \text{exterior angle} = 180^\circ – 135^\circ = 45^\circ \]Step 6: Number of sides \( n = \frac{360^\circ}{\text{exterior angle}} = \frac{360^\circ}{45^\circ} = 8 \).
Answer: The polygon has 8 sides.

Q9: The measure of each interior angle of a regular polygon is five times the measure of its exterior angle. Find:

i. Measure of each interior angle
ii. Measure of each exterior angle
iii. Number of sides in the polygon

Step 1: Let each exterior angle be \( x^\circ \). Then each interior angle is \( 5x^\circ \).
Step 2: Interior and exterior angles are supplementary: \[ x + 5x = 180^\circ \Rightarrow 6x = 180^\circ \]Step 3: Solve for \( x \): \[ x = \frac{180^\circ}{6} = 30^\circ \]Step 4: Find the interior angle: \[ 5x = 5 \times 30^\circ = 150^\circ \]Step 5: Number of sides \( n \): \[ n = \frac{360^\circ}{\text{exterior angle}} = \frac{360^\circ}{30^\circ} = 12 \]Answer: i. Each interior angle = 150°
ii. Each exterior angle = 30°
iii. Number of sides = 12

Q10: The ratio between the interior angle and the exterior angle of a regular polygon is 2 : 1. Find:

i. Each exterior angle of the polygon

Step 1: Let the interior angle be \( 2x^\circ \) and the exterior angle be \( x^\circ \).
Step 2: Since interior and exterior angles are supplementary: \[ 2x + x = 180^\circ \Rightarrow 3x = 180^\circ \]Step 3: Solve for \( x \): \[ x = \frac{180^\circ}{3} = 60^\circ \]Answer: Each exterior angle = 60°

ii. Number of sides in the polygon

Step 4: Exterior angle \( = 60^\circ \).
Step 5: Number of sides \( n \): \[ n = \frac{360^\circ}{60^\circ} = 6 \]Answer: Number of sides = 6

Q11: The ratio between the exterior angle and the interior angle of a regular polygon is 1 : 4. Find the number of sides in the polygon.

Step 1: Let the exterior angle be \( x^\circ \). Then the interior angle is \( 4x^\circ \).
Step 2: Since interior and exterior angles are supplementary: \[ x + 4x = 180^\circ \Rightarrow 5x = 180^\circ \]Step 3: Solve for \( x \): \[ x = \frac{180^\circ}{5} = 36^\circ \]Step 4: Number of sides \( n \): \[ n = \frac{360^\circ}{\text{exterior angle}} = \frac{360^\circ}{36^\circ} = 10 \]Answer: The polygon has 10 sides.

Q12: The sum of interior angles of a regular polygon is twice the sum of its exterior angles. Find the number of sides of the polygon.

Step 1: Let the number of sides be \( n \).
Step 2: Sum of interior angles of a polygon: \[ (n – 2) \times 180^\circ \]Step 3: Sum of exterior angles of any polygon is always \( 360^\circ \).
Step 4: Given that sum of interior angles is twice the sum of exterior angles: \[ (n – 2) \times 180^\circ = 2 \times 360^\circ \]Step 5: Simplify and solve for \( n \): \[ (n – 2) \times 180^\circ = 720^\circ \\ n – 2 = \frac{720^\circ}{180^\circ} = 4 \\ n = 4 + 2 = 6 \]Answer: The polygon has 6 sides.

Q13: AB, BC and CD are three consecutive sides of a regular polygon. If angle BAC = 20°, find:

i. its each interior angle

Step 1: Analyze the triangle formed by the sides of the polygon.
In a regular polygon, all sides are equal. Therefore, AB = BC.
In triangle ABC, since two sides are equal (AB = BC), it is an isosceles triangle.
The angles opposite to equal sides are equal, so ∠BCA = ∠BAC = 20°.
Step 2: Find the interior angle of the polygon.
The angle ∠ABC is one of the interior angles of this regular polygon.
In ΔABC, the sum of angles is 180°:
∠ABC + ∠BAC + ∠BCA = 180°
∠ABC + 20° + 20° = 180°
∠ABC + 40° = 180°
∠ABC = 180° – 40° = 140°
Each interior angle = 140°
Answer: 140°

ii. its each exterior angle

Step 3: Calculate the exterior angle.
Exterior angle = 180° – Interior angle
Exterior angle = 180° – 140° = 40°
Answer: 40°

iii. the number of sides in the polygon

Step 4: Find the number of sides (n) of the polygon.
Formula: Number of sides (n) = 360° / Exterior angle
n = 360° / 40°
n = 9
Answer: 9

Q14: Two alternate sides of a regular polygon, when produced, meet at right angle. Calculate the number of sides in the polygon.

Step 1: Let the number of sides in the polygon be \( n \).
Step 2: The exterior angle of a regular polygon is: \[ \text{Exterior angle} = \frac{360^\circ}{n} \]Step 3: Two alternate sides mean they are separated by one side.
The angle between these two sides when produced equals twice the exterior angle: \[ \text{Angle between alternate sides} = 2 \times \text{Exterior angle} = 2 \times \frac{360^\circ}{n} = \frac{720^\circ}{n} \]Step 4: Given that these produced sides meet at right angle, so: \[ \frac{720^\circ}{n} = 90^\circ \]Step 5: Solve for \( n \): \[ n = \frac{720^\circ}{90^\circ} = 8 \]Answer: The polygon has 8 sides.

Q15: In a regular pentagon ABCDE, draw a diagonal BE and then find the measure of:

i. ∠BAE

A pentagon has n = 5 sides.
Sum of interior angles = (n – 2) 180° = (5 – 2) 180° = 3 180° = 540°
Each interior angle = 540° / 5 = 108°
Therefore, ∠BAE = 108°
Answer: 108°

ii. ∠ABE

In triangle ABE, side AB = side AE (since it is a regular pentagon).
Thus, ΔABE is an isosceles triangle, which means ∠ABE = ∠AEB.
In ΔABE:
∠BAE + ∠ABE + ∠AEB = 180°
108° + 2 ∠ABE = 180°
2 ∠ABE = 180° – 108° = 72°
∠ABE = 72° / 2 = 36°
Answer: 36°

iii. ∠BED

We know that the full interior angle ∠AED = 108°.
From the figure, ∠AED = ∠AEB + ∠BED.
We found in Step 2 that ∠AEB = 36°.
108° = 36° + ∠BED
∠BED = 108° – 36° = 72°
Answer: 72°

Q16: The ratio between the number of sides of two regular polygons is 3 : 4 and the ratio between the sum of their interior angles is 2 : 3. Find the number of sides in each polygon.

Step 1: Let the number of sides of the two polygons be \(3x\) and \(4x\).
Step 2: Sum of interior angles of a polygon with \(n\) sides is: \[ (n – 2) \times 180^\circ \]Step 3: Ratio of sum of interior angles: \[ \frac{(3x – 2) \times 180^\circ}{(4x – 2) \times 180^\circ} = \frac{2}{3} \] Cancel \(180^\circ\): \[ \frac{3x – 2}{4x – 2} = \frac{2}{3} \]Step 4: Cross multiply and solve for \( x \): \[ 3(3x – 2) = 2(4x – 2) \\ 9x – 6 = 8x – 4 \\ 9x – 8x = -4 + 6 \\ x = 2 \]Step 5: Find number of sides: \[ \text{Polygon 1 sides} = 3x = 3 \times 2 = 6 \\ \text{Polygon 2 sides} = 4x = 4 \times 2 = 8 \]Answer: The first polygon has 6 sides, and the second polygon has 8 sides.

Q17: Three of the exterior angles of a hexagon are 40°, 51° and 86°. If each of the remaining exterior angles is \( x^\circ \), find the value of \( x \).

Step 1: Sum of all exterior angles of any polygon is \( 360^\circ \).
Step 2: Given three exterior angles: 40°, 51°, and 86°.
Remaining three exterior angles each measure \( x^\circ \).
Step 3: Write the equation for sum of exterior angles: \[ 40^\circ + 51^\circ + 86^\circ + 3x = 360^\circ \]Step 4: Calculate sum of known angles: \[ 40^\circ + 51^\circ + 86^\circ = 177^\circ \]Step 5: Substitute and solve for \( x \): \[ 177^\circ + 3x = 360^\circ \Rightarrow 3x = 360^\circ – 177^\circ = 183^\circ \\ x = \frac{183^\circ}{3} = 61^\circ \]Answer: Each of the remaining exterior angles is 61°.

Q18: Calculate the number of sides of a regular polygon if:

i. Its interior angle is five times its exterior angle.

Step 1: Let exterior angle = \( x^\circ \), interior angle = \( 5x^\circ \).
Since interior and exterior angles are supplementary: \[ x + 5x = 180^\circ \Rightarrow 6x = 180^\circ \Rightarrow x = 30^\circ \]Step 2: Number of sides \( n = \frac{360^\circ}{x} = \frac{360^\circ}{30^\circ} = 12 \).
Answer: Number of sides = 12

ii. The ratio between its exterior angle and interior angle is 2 : 7.

Step 1: Let exterior angle = \( 2x^\circ \), interior angle = \( 7x^\circ \).
Since interior and exterior angles are supplementary: \[ 2x + 7x = 180^\circ \Rightarrow 9x = 180^\circ \Rightarrow x = 20^\circ \]Step 2: Exterior angle \( = 2x = 40^\circ \).
Number of sides \( n = \frac{360^\circ}{40^\circ} = 9 \).
Answer: Number of sides = 9

iii. Its exterior angle exceeds its interior angle by 60°.

Step 1: Let interior angle = \( x^\circ \), exterior angle = \( x + 60^\circ \).
Since interior and exterior angles are supplementary: \[ x + (x + 60^\circ) = 180^\circ \Rightarrow 2x + 60^\circ = 180^\circ \Rightarrow 2x = 120^\circ \Rightarrow x = 60^\circ \]Step 2: Exterior angle \( = 60^\circ + 60^\circ = 120^\circ \).
Step 3: Number of sides \( n = \frac{360^\circ}{120^\circ} = 3 \).
Answer: Number of sides = 3