Proving using induction. Polygons and (n-2)*(180)

Question

Show that if n > 2 points are spread round a circle and are connected in order to form a polygon, then the interior angles
of the polygon add up to (n − 2) × 180◦

Basis: let n=3

3>2 which is true 

so 

when n=3 there are 3 points on the circumference of the circle, (3-2)=1 indicates how many triangles it produces.

(n-2)*(180) is the sum of the angles 

(3-2)*(180) = 180 because there is only one triangle.

 

How do i prove this using induction

Answer 1

 

Assume it is true for n=k,  k≥3.

 

Now let n=k+1.   Draw the k+1 sided polygon.   Now connect vertex k with vertex 1 to form a triangle (vertices k, k+1, and 1  form the three vertices of the triangle).   The polygon excluding this triangle is a k-sided polygon and we know from the induction hypothesis that  the sum of the interior angles is 180*(k-2) for this part.   Now add back that one triangle, that brings the sum of interior angles to:

 180+180(k-2) =

 180(k-1) =

 180((k+1)-2)

 

This completes the proof (if true for n=k, then it follows it is true for n=k+1).