Chapter 11.7, Problem 16PPS

To determine

Prove this formula for n = 3 using mathematical induction and geometry.

Answer to Problem 16PPS

Summing these we get $(n+1-2)·{180}^0$ , and the theorem is proved.

Explanation of Solution

Given:

According to the interior angle sum formula, if a convex polygon has n sides, then sum of the measures of the interior angles of a polygon equals 180 (n − 2). Prove this formula for n = 3 using mathematical induction and geometry. Concept Used:

Step 1: When $n=3,180(3-2)=180.$

When a polygon has 3 sides or 3 vertices, there are 180 degrees. The statement is true for n = 3

Step 2: Assume that the statement is true for $n\ge 3.$

Step 3: Consider a convex polygon with n + 1 vertices. Since $n+1\ge 4,$ if we take one vertex a there is another vertex y such that there is one vertex between x and y in one direction, and n − 2 in the other direction. Join x and y by a new edge, dividing our original polygon into two polygons. The new polygons` interior angles summed together make up the sum of the original polygon`s interior angles. One of the new polygons is a triangle and the other a polygon of n vertices (all but the one isolated between x and y). The triangle has interior angle sum 180 degrees, and by the inductive hypothesis the other polygon has interior angle sum $(n-2)·{180}^0$

Summing these we get $(n+1-2)·{180}^0$ , and the theorem is proved.

Thus, summing these we get $(n+1-2)·{180}^0$ , and the theorem is proved.