To find the sum of the interior angles of an irregular polygon with [tex]\( n = 13 \)[/tex] sides, where two of the interior angles are right angles (90 degrees each), and the exterior angles of each of the remaining angles are [tex]\( 30^\circ \)[/tex], follow these steps:
1. Identify the total number of sides and the given information:
- The polygon has [tex]\( n = 13 \)[/tex] sides.
- Two interior angles are right angles ([tex]\( 90^\circ \)[/tex] each).
- The exterior angle of each of the remaining [tex]\( 13 - 2 = 11 \)[/tex] angles is [tex]\( 30^\circ \)[/tex].
2. Calculate the sum of the two right angles:
The total for the two right angles:
[tex]\[
2 \times 90^\circ = 180^\circ
\][/tex]
3. Relate the exterior angles to the interior angles:
The relationship between an exterior angle and its corresponding interior angle is:
[tex]\[
\text{interior angle} = 180^\circ - \text{exterior angle}
\][/tex]
Therefore, for each of the remaining 11 angles with an exterior angle of [tex]\( 30^\circ \)[/tex]:
[tex]\[
\text{interior angle} = 180^\circ - 30^\circ = 150^\circ
\][/tex]
4. Calculate the sum of the remaining interior angles:
Since there are 11 such angles, the sum of these angles is:
[tex]\[
11 \times 150^\circ = 1650^\circ
\][/tex]
5. Calculate the total sum of all interior angles:
Adding the sum of the two right angles and the remaining interior angles:
[tex]\[
\text{Sum of interior angles} = 180^\circ + 1650^\circ = 1830^\circ
\][/tex]
Therefore, for a polygon with [tex]\( n = 13 \)[/tex] sides, given the specific interior and exterior angles, the sum of the interior angles is:
[tex]\[
1830^\circ
\][/tex]
