124 Consider a (k + I)-gon. Let P be a vertex and Q, R the vertices adjacent to P. Join QR as shown on the right. Since k ?: 3, the line Q R separates the figure into two distinct pieces, triangle PQ R and the k-gon fonned by QR and all sides of the original (k + I)-gon except PQ and P R. By the induction hypothesis, the sum of the interior angles of this new k-gon equals (k -2) 1800• Also, triangle PQR has sum of interior angles equal to 1800• Solutions to Exercises The sum of the angles of the (k + 1 )-gon is the sum of the angles in the k-gon and in the triangle, that is, (k -2) 1800 + 1800 = (k -1) 1800, as desired. By the Principle of Mathematical Induction, the result is true for all n ?: 3. 29. We prove this by induction on n, the number of straight lines drawn. If n = 1, we have two regions which, if colored with different colors, gives a proper coloring using just two colors. (By "proper," we mean that bordering countries have different colors.) Now suppose that k ?: 1 and the statement is true for n = k; that is, suppose that a map made by drawing k
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