Absolutely! I'd be happy to provide a detailed, step-by-step solution for these questions.
### Exterior Angles of Regular Polygons
The exterior angle of a regular polygon is given by the formula:
[tex]\[ \text{Exterior Angle} = \frac{360^\circ}{\text{number of sides}} \][/tex]
Let's find the exterior angles for each polygon:
#### a) 10 Sides
[tex]\[ \text{Exterior Angle} = \frac{360^\circ}{10} = 36.0^\circ \][/tex]
#### b) 6 Sides
[tex]\[ \text{Exterior Angle} = \frac{360^\circ}{6} = 60.0^\circ \][/tex]
#### e) 12 Sides
[tex]\[ \text{Exterior Angle} = \frac{360^\circ}{12} = 30.0^\circ \][/tex]
#### f) 15 Sides
[tex]\[ \text{Exterior Angle} = \frac{360^\circ}{15} = 24.0^\circ \][/tex]
#### i) 20 Sides
[tex]\[ \text{Exterior Angle} = \frac{360^\circ}{20} = 18.0^\circ \][/tex]
#### j) 4 Sides
[tex]\[ \text{Exterior Angle} = \frac{360^\circ}{4} = 90.0^\circ \][/tex]
#### d) 8 Sides
[tex]\[ \text{Exterior Angle} = \frac{360^\circ}{8} = 45.0^\circ \][/tex]
#### c) 9 Sides
[tex]\[ \text{Exterior Angle} = \frac{360^\circ}{9} = 40.0^\circ \][/tex]
#### g) 18 Sides
[tex]\[ \text{Exterior Angle} = \frac{360^\circ}{18} = 20.0^\circ \][/tex]
#### h) 16 Sides
[tex]\[ \text{Exterior Angle} = \frac{360^\circ}{16} = 22.5^\circ \][/tex]
#### k) 36 Sides
[tex]\[ \text{Exterior Angle} = \frac{360^\circ}{36} = 10.0^\circ \][/tex]
### Sum of Interior Angles of a Regular Pentagon
To find the sum of the interior angles of a regular pentagon (which has 5 sides), we use the formula:
[tex]\[ \text{Sum of Interior Angles} = (\text{number of sides} - 2) \times 180^\circ \][/tex]
For a pentagon:
[tex]\[ \text{Sum of Interior Angles} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ \][/tex]
Thus, the sum of the interior angles of a regular pentagon is [tex]\( 540^\circ \)[/tex].
In summary:
- The exterior angles for polygons with 10, 6, 12, 15, 20, 4, 8, 9, 18, 16, and 36 sides are 36.0°, 60.0°, 30.0°, 24.0°, 18.0°, 90.0°, 45.0°, 40.0°, 20.0°, 22.5°, and 10.0°, respectively.
- The sum of the interior angles of a regular pentagon is 540°.
