Question

PLEASE HELP QUICK! The following polygons are given. All of the polygons are regular polygons.

Answer 1

1. Convex 15-gon, Convex 18-gon, Convex 45-gon

2. Increases

3. Decreases

4. Increases

5. Stays the same

Explanation:

The formula for finding the measure of an interior angle, θ, of a polygon with 'n' sides is given as follows;

`\theta = ((n - 2) * 180)/(n)`

1. Polygons that have interior angles with whole numbers are factors of 180

The factors of 180 are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, and 180

Therefore, from the options, the polygon that have whole number interior angles are;

15, 18, and 45

Which gives;

Convex 15-gon

θ₁₅ = (15 - 2) × 180°/15 = 156°

Convex 18-gon

θ₁₈ = (18 - 2) × 180°/18 = 160°

Convex 45-gon

θ₄₅ = (45 - 2) × 180°/45 = 172°

2. From the equation of the interior angles of a polygon,
`\theta = ((n - 2) * 180)/(n)`, we have;

As n increases, (n - 2)/n increases towards 1

Therefore, as the number of sides 'n' of a polygon increases, the interior angle of the polygon increases

3. Given that the sum of the interior and exterior angles of a polygon is 180°, as the interior angles increases as the number of sides increases, the exterior angle decreases

4. The total sum of the angles increases as the number of sides of the polygon increases and decreases as the number of sides of the polygon decreases

5. The sum of the exterior angles of a polygon is given as follows

`n * (180^(\circ) - \theta) = n * \left(180^(\circ) - ((n - 2) * 180^(\circ))/(n) \right) = n\cdot 180^(\circ) - n\cdot 180^(\circ) + 360^(\circ)`

`n * (180^(\circ) - \theta) = 360^(\circ)`

Therefore, total sum of the exterior angles of a regular polygon is always (remains the same) 360°.