Question

a polygon has "n" sides. Two of ti's angles are right angles. Each of the remaining angles is equal to 150 degrees. Calculate "n"

Answer 1

To determine the number of sides \( n \) in the polygon, we can use the formula for the sum of the interior angles of a polygon, which is given by:

\[
S = (n - 2) \times 180^\circ
\]

According to the problem, the polygon has two right angles (which are each \( 90^\circ \)) and the remaining angles are each \( 150^\circ \). Let's denote the number of remaining angles as \( n - 2 \) since there are two angles that are \( 90^\circ \).

1. The total sum of the interior angles can also be expressed as the sum of all the angles:
- Total from the right angles: \( 2 \times 90^\circ = 180^\circ \)
- Total from the remaining angles: \( (n - 2) \times 150^\circ \)

Thus, we can write the equation for the sum of the angles:

\[
180^\circ + (n - 2) \times 150^\circ = (n - 2) \times 180^\circ
\]

2. Now, let's simplify this equation:

- Expanding both sides:

\[
180 + 150(n - 2) = 180(n - 2)
\]

- Distributing on both sides:

\[
180 + 150n - 300 = 180n - 360
\]

- Simplifying gives:

\[
150n - 120 = 180n - 360
\]

3. Now, let's combine like terms:

\[
-120 + 360 = 180n - 150n
\]

\[
240 = 30n
\]

4. Finally, solving for \( n \):

\[
n = \frac{240}{30} = 8
\]

Thus, the number of sides \( n \) of the polygon is

\[
\boxed{8}
\]