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Given:
[tex] n = 13 [/tex]

An irregular polygon has [tex]\( n \)[/tex] sides. Its interior angles are such that 2 of them are right angles, while the exterior of each of the remaining angles is [tex]\( 30^{\circ} \)[/tex]. Find the value of [tex]\( n \)[/tex] and hence the sum of the interior angles of the polygon.

Sagot :

GinnyAnswer
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]
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