Question

Prove with induction that all ​convex​ polygons with n≥3 sides have interior angles that add up to (n-2)·180 degrees. You may assume that a triangle has interior angles that add up to 180 degrees.

Answer 1

Hence by induction proved for all natural numbers n.

Explanation:

we are to Prove with induction that all ​convex​ polygons with n≥3 sides have interior angles that add up to (n-2)·180 degrees.

Starting from triangle we can assume that angles of a triangle add up to 180

Imagine one side say AB. From A and B two lines are drawn to meet at D

Now BADC is a quadrilateral. The sum of angles of a quadrilateral would be sum of angles of two triangles namely ABC and BDC. hence these add up to 360.

Thus when we make n from 3 to 4 this is true.

Let us assume for n sides sum of angles is (n-2)180 degrees. Take one side vertices and draw two lines so that the polygon is n+1 sided. Now the total angles would be the sum of angles of original polygon+angles of new triangle = (n-2)180+1 = (n+1-2)180

Thus if true for n it is true for n+1. Already true for 3 and 4.

Hence by induction proved for all natural numbers n.