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"

1 answer

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:

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

  1. 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:

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

  1. Now, let's combine like terms:

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

\[ 240 = 30n \]

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

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

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

\[ \boxed{8} \]