Remember: Apolygon is called a convex if each of it\'sinterior angles is less th
ID: 2937644 • Letter: R
Question
Remember: Apolygon is called a convex if each of it'sinterior angles is less than . Recall that a polygon isregular if all of the sides have the same length, and all of theinterior angles are equal. a) Prove that for a convex polygon with n vertices, the sum ofthe interior angles is equal to (n-2). Remember: Apolygon is called a convex if each of it'sinterior angles is less than . Recall that a polygon isregular if all of the sides have the same length, and all of theinterior angles are equal. a) Prove that for a convex polygon with n vertices, the sum ofthe interior angles is equal to (n-2).Explanation / Answer
Let P be a point inside the polygon. Join all the vertices ofthe polygon to this point. This way, we divide the polygon into n triangles. The sum of the angles of a triangle is. Since there are n triangles, the sum of all theangles of all these triangles is n. On the other hand,this sum equals the sum of all the angles of the polygon, plus2 (which is the total contribution of all the angles atP). Thus, the sum of all the angles of the polygon = n- 2 = (n-2).