Answer :
To find the size of each interior angle of a regular polygon, we use the formula for the measure of an interior angle:
[tex]\[ \text{Interior Angle} = \frac{180(n - 2)}{n} \][/tex]
where [tex]\( n \)[/tex] is the number of sides of the polygon.
Let's solve each part step-by-step:
### a) For a polygon with 4 sides (a quadrilateral):
- The number of sides [tex]\( n = 4 \)[/tex]
- Applying the formula:
[tex]\[ \text{Interior Angle} = \frac{180(4 - 2)}{4} = \frac{180 \times 2}{4} = \frac{360}{4} = 90^\circ \][/tex]
So, each interior angle of a regular quadrilateral is [tex]\( 90^\circ \)[/tex].
### b) For a polygon with 5 sides (a pentagon):
- The number of sides [tex]\( n = 5 \)[/tex]
- Applying the formula:
[tex]\[ \text{Interior Angle} = \frac{180(5 - 2)}{5} = \frac{180 \times 3}{5} = \frac{540}{5} = 108^\circ \][/tex]
So, each interior angle of a regular pentagon is [tex]\( 108^\circ \)[/tex].
### c) For a polygon with 8 sides (an octagon):
- The number of sides [tex]\( n = 8 \)[/tex]
- Applying the formula:
[tex]\[ \text{Interior Angle} = \frac{180(8 - 2)}{8} = \frac{180 \times 6}{8} = \frac{1080}{8} = 135^\circ \][/tex]
So, each interior angle of a regular octagon is [tex]\( 135^\circ \)[/tex].
### d) For a polygon with 10 sides (a decagon):
- The number of sides [tex]\( n = 10 \)[/tex]
- Applying the formula:
[tex]\[ \text{Interior Angle} = \frac{180(10 - 2)}{10} = \frac{180 \times 8}{10} = \frac{1440}{10} = 144^\circ \][/tex]
So, each interior angle of a regular decagon is [tex]\( 144^\circ \)[/tex].
In summary,
- Each interior angle of a regular quadrilateral is [tex]\( 90^\circ \)[/tex].
- Each interior angle of a regular pentagon is [tex]\( 108^\circ \)[/tex].
- Each interior angle of a regular octagon is [tex]\( 135^\circ \)[/tex].
- Each interior angle of a regular decagon is [tex]\( 144^\circ \)[/tex].
[tex]\[ \text{Interior Angle} = \frac{180(n - 2)}{n} \][/tex]
where [tex]\( n \)[/tex] is the number of sides of the polygon.
Let's solve each part step-by-step:
### a) For a polygon with 4 sides (a quadrilateral):
- The number of sides [tex]\( n = 4 \)[/tex]
- Applying the formula:
[tex]\[ \text{Interior Angle} = \frac{180(4 - 2)}{4} = \frac{180 \times 2}{4} = \frac{360}{4} = 90^\circ \][/tex]
So, each interior angle of a regular quadrilateral is [tex]\( 90^\circ \)[/tex].
### b) For a polygon with 5 sides (a pentagon):
- The number of sides [tex]\( n = 5 \)[/tex]
- Applying the formula:
[tex]\[ \text{Interior Angle} = \frac{180(5 - 2)}{5} = \frac{180 \times 3}{5} = \frac{540}{5} = 108^\circ \][/tex]
So, each interior angle of a regular pentagon is [tex]\( 108^\circ \)[/tex].
### c) For a polygon with 8 sides (an octagon):
- The number of sides [tex]\( n = 8 \)[/tex]
- Applying the formula:
[tex]\[ \text{Interior Angle} = \frac{180(8 - 2)}{8} = \frac{180 \times 6}{8} = \frac{1080}{8} = 135^\circ \][/tex]
So, each interior angle of a regular octagon is [tex]\( 135^\circ \)[/tex].
### d) For a polygon with 10 sides (a decagon):
- The number of sides [tex]\( n = 10 \)[/tex]
- Applying the formula:
[tex]\[ \text{Interior Angle} = \frac{180(10 - 2)}{10} = \frac{180 \times 8}{10} = \frac{1440}{10} = 144^\circ \][/tex]
So, each interior angle of a regular decagon is [tex]\( 144^\circ \)[/tex].
In summary,
- Each interior angle of a regular quadrilateral is [tex]\( 90^\circ \)[/tex].
- Each interior angle of a regular pentagon is [tex]\( 108^\circ \)[/tex].
- Each interior angle of a regular octagon is [tex]\( 135^\circ \)[/tex].
- Each interior angle of a regular decagon is [tex]\( 144^\circ \)[/tex].
To find the size of each interior angle of a regular polygon, we can use the formula:
Interior angle = (n - 2) × 180° / n
where n is the number of sides.
Let's apply this formula to each case:
a) 4 sides (quadrilateral):
Interior angle = (4 - 2) × 180° / 4 = 2 × 180° / 4 = 90°
b) 5 sides (pentagon):
Interior angle = (5 - 2) × 180° / 5 = 3 × 180° / 5 = 108°
c) 8 sides (octagon):
Interior angle = (8 - 2) × 180° / 8 = 6 × 180° / 8 = 135°
d) 10 sides (decagon):
Interior angle = (10 - 2) × 180° / 10 = 8 × 180° / 10 = 144°
So, the size of each interior angle is:
- 90° for a quadrilateral (4 sides)
- 108° for a pentagon (5 sides)
- 135° for an octagon (8 sides)
- 144° for a decagon (10 sides)
Interior angle = (n - 2) × 180° / n
where n is the number of sides.
Let's apply this formula to each case:
a) 4 sides (quadrilateral):
Interior angle = (4 - 2) × 180° / 4 = 2 × 180° / 4 = 90°
b) 5 sides (pentagon):
Interior angle = (5 - 2) × 180° / 5 = 3 × 180° / 5 = 108°
c) 8 sides (octagon):
Interior angle = (8 - 2) × 180° / 8 = 6 × 180° / 8 = 135°
d) 10 sides (decagon):
Interior angle = (10 - 2) × 180° / 10 = 8 × 180° / 10 = 144°
So, the size of each interior angle is:
- 90° for a quadrilateral (4 sides)
- 108° for a pentagon (5 sides)
- 135° for an octagon (8 sides)
- 144° for a decagon (10 sides)
