Question 1183295
A polygon has two interior angles of 120 degrees each and the others are 150 degrees.Calculate a. The number of sides of the polygon b. The sum of interior angles
<pre>Since 2 interior angles measure 120<sup>o</sup> each, then 2 exterior angles measure 180<sup>o</sup> - 120<sup>o</sup> = 60<sup>o</sup> each
Let number of angles (exterior/interior), be n
With 2 angles known, remaining angles = n - 2
Since remaining interior angles measure 150<sup>o</sup> each, remaining exterior angles measure 180<sup>o</sup> - 150<sup>o</sup> = 30<sup>o</sup> 
each. So, remaining exterior angles measure a total of 30(n - 2) = 30n - 60

With all exterior angles measuring 120<sup>o</sup> and (30n - 60)<sup>o</sup>, and since the exterior angles of any polygon sum
to 360<sup>o</sup>, we now have: 120 + 30n - 60 = 360
60 + 30n = 360
30n = 300
Number of sides, or {{{highlight_green(matrix(1,5, n, "=", 300/30, "=", 10))}}}

Sum of interior angles: {{{highlight_green(matrix(1,7, 180(n - 2), "=", 180(10 - 2), "=", 180(8), "=", "1,440"^o))}}}

<b><u>OR</b></u>

Two (2) interior angles measure 120<sup>o</sup> each, so 2 interior angles measure 2(120)<sup>o</sup>

Let number of angles (exterior/interior), be n
With 2 interior angles known, remaining angles = n - 2
Since remaining interior angles measure 150<sup>o</sup> each, remaining interior angles measure 180(n - 2)<sup>o</sup>

With all interior angles measuring 2(120)<sup>o</sup> and 150(n - 2)<sup>o</sup>, and since the interior angles of any polygon sum
to 180(n - 2)<sup>o</sup>, we now have: 2(120) + 150(n - 2) = 180(n - 2)
240 = 180(n - 2) - 150(n - 2)
240 = 30(n - 2)
{{{matrix(2,3, 240/30, "=", n - 2, 8, "=", n - 2))}}}
Number of sides, or {{{highlight_green(matrix(2,3, n, "=", 8 + 2, n, "=", 10))}}}

Sum of interior angles: {{{highlight_green(matrix(1,7, 180(n - 2), "=", 180(10 - 2), "=", 180(8), "=", "1,440"^o))}}}</pre>