Prove the two corollaries given by products of proposition I

Prove the two corollaries given by products of proposition I - 32 The sum of the interior angles of a convex rectilinear figure equals twice as many angles as the figure has sides, less four The sum of the exterior angles of any convex rectilinear figure together equal four right angles.

Solution

1. For n sided polygon we can divide it in n traingles by taking any point P inside polygon and drawing straight lines to eah vertex.
Now angle of these traingles will be 2*90 (twice the right angle). Combining all these n traingles we have 2n*90 (2n times right angle) which include interior angles + all angles at that point P.
sum of all angle at P = 4*90 (4 times right angle)
Hence sum of interior angles = (2n-4) right angles

2. Divide a polygon of n sides into triangles such that these traingles are made by non-intersecting diagonals between the vertices of the polygon. We will end up in getting n-2 such traingles for n sided polygon.
Hence internal angle = (n-2)*180 = 180n - 360
Also, n sided polygon has n vertices which will make angle = 180*n
Hence sum of exterior angle = 180n - (180n - 360) = 360 (4 right angles)