Use an indirect proof to establish the following theorem: Given:

A concave polygon can have more than one reflex angle. Table 2.3 shows some special names for polygons with fixed numbers of sides. TABLE 2.3 Polygon Number of Sides Polygon Number of Sides Triangle 3 Heptagon 7 Quadrilateral 4 Octagon 8 Pentagon 5 Nonagon 9 Hexagon 6 Decagon 10 With Venn Diagrams, the set of all objects under consideration is called the uni- verse. If P ⫽ {all polygons} is the universe, then we can describe sets T ⫽ {triangles} and Q ⫽ {quadrilaterals} as subsets that lie within universe P. Sets T and Q are described as disjoint because they have no elements in common. See Figure 2.31. T P Q Figure 2.31 2 1 3 F E D C B A G Figure 2.32 DIAGONALS OF A POLYGON A diagonal of a polygon is a line segment that joins two nonconsecutive vertices. Figure 2.32 shows heptagon ABCDEFG for which , , and are some of the interior angles and , , and are some of the exterior angles. , , and are some of the sides of the heptagon, because these join consecutive ver- tices. Because a diagonal joins nonconsecutive vertices of ABCDEFG, , , and are among the many diagonals of the polygon. Table 2.4 illustrates polygons by numbers of sides and the corresponding total number of diagonals for each type. When the number of sides of a polygon is small, we can list all diagonals by name. For pentagon ABCDE of Table 2.4, we see diagonals , , , , and —a total of five. As the number of sides increases, it becomes more difficult to count all the CE BE BD AD AC AE AD AC CD BC AB ∠3 ∠2 ∠1 ∠BCD ∠B ∠GAB TABLE 2.4 Triangle Quadrilateral Pentagon Hexagon 3 sides 4 sides 5 sides 6 sides 0 diagonals 2 diagonals 5 diagonals 9 diagonals P Q M N B C D E A O L M N Q P R T S diagonals. In such a case, the formula of Theorem 2.5.1 is most convenient to use. Al- though this theorem is given without proof, Exercise 39 of this section provides some insight for the proof. EXAMPLE 1 Use Theorem 2.5.1 to find the number of diagonals for any pentagon. Solution To use the formula of Theorem 2.5.1, we note that n ⫽ 5 in a pentagon. Then . 쮿 D = 55 - 3 2 = 52 2 = 5 Reminder The sum of the interior angles of a triangle is 180°. Exs. 1–5 SUM OF THE INTERIOR ANGLES OF A POLYGON The following theorem provides the formula for the sum of the interior angles of any polygon. The sum S of the measures of the interior angles of a polygon with n sides is given by . Note that for any polygon. n 7 2 S = n - 2 180° THEOREM 2.5.2 Let us consider an informal proof of Theorem 2.5.2 for the special case of a penta- gon. The proof would change for a polygon of a different number of sides but only by the number of triangles into which the polygon can be separated. Although Theorem 2.5.2 is also true for concave polygons, we consider the proof only for the case of the convex polygon. Proof Consider the pentagon ABCDE in Figure 2.33 with auxiliary segments diagonals from one vertex as shown. With angles marked as shown in triangles ABC, ACD, and ADE, m 1 ⫹ m 2 ⫹ m 3 ⫽ 180⬚ m 6 ⫹ m 5 ⫹ m 4 ⫽ 180⬚ m 8 ⫹ m 9 ⫹ m 7 ⫽ 180⬚ m E ⫹ m A ⫹ m D ⫹ m B ⫹ m C ⫽ 540⬚ adding For pentagon ABCDE, in which , the sum of the measures of the interior angles is , which equals 540°. When drawing diagonals from one vertex of a polygon of n sides, we always form triangles. The sum of the measures of the interior angles always equals . n - 2 180° n - 2 5 - 2 180° n = 5 ∠ ∠ ∠ ∠ ∠ ∠ ∠ ∠ ∠ ∠ ∠ ∠ ∠ ∠ E D C B 2

3 4

5 7 8 1 6 9 A Figure 2.33 The total number of diagonals D in a polygon of n sides is given by the formula . D = n n - 3 2 THEOREM 2.5.1 Theorem 2.5.1 reaffirms the fact that a triangle has no diagonals; when , D = 33 - 3 2 = 0. n = 3 쮿