8.2 Polygons 465 One of the most important polygons is the triangle. The sum of the measures of the interior angles of a triangle is ° 180 . To illustrate, consider triangle ABC given in Fig. 8.18. The triangle is formed by drawing two transversals through two parallel lines l1 and l2 with the two transversals intersecting at a point on l .1 In Fig. 8.18, notice that A and ′A are corresponding angles. Recall from Section 8.1 that corresponding angles are equal, so = ′ m A m A. Also, C and ′C are corresponding angles; therefore, = ′ m C m C. Next, we notice that B and ′B are vertical angles. In Section 8.1, we learned that vertical angles are equal; therefore, = ′ m B m B. Fig. 8.18 shows that ′ ′ A B , , and ′C form a straight angle; therefore, ′ + ′ + ′ = m A m B m C 180°. Since = ′ = ′ m A m A m B m B , , and = ′ m C m C , we can reason that + + = m A m B m C 180°. This discussion illustrates that the sum of the interior angles of a triangle is ° 180 . Consider the quadrilateral ABCD (Fig. 8.19a). Drawing a straight line segment between any two vertices forms two triangles. Since the sum of the measures of the angles of a triangle is ° 180 , the sum of the measures of the interior angles of a quadrilateral is ⋅ 2 180°, or ° 360 . Now let’s examine a pentagon (Fig. 8.19b). We can draw two straight line segments to form three triangles. Thus, the sum of the measures of the interior angles of a five-sided figure is ⋅ 3 180°, or ° 540 . Figure 8.19(c) shows that four triangles can be drawn in a six-sided figure. Table 8.2 summarizes this information. D B C A (a) (b) (c) Figure 8.19 Sides and Angles of Polygons A polygon is a closed figure in a plane determined by three or more straight line segments. Examples of polygons are given in Fig. 8.17. (a) (b) (c) (d) Figure 8.17 A' B' B C A l2 l1 C' Figure 8.18 Table 8.1 Number of Sides Name Number of Sides Name 3 Triangle 8 Octagon 4 Quadrilateral 9 Nonagon 5 Pentagon 10 Decagon 6 Hexagon 12 Dodecagon 7 Heptagon 20 Icosagon The straight line segments that form the polygon are called its sides, and a point where two sides meet is called a vertex (plural, vertices). The union of the sides of a polygon and its interior is called a polygonal region. A regular polygon is one whose sides are all the same length and whose interior angles all have the same measure. Fig. 8.17(b) and (d) are regular polygons. Polygons are named according to their number of sides. The names of some polygons are given in Table 8.1. Learning Catalytics Keyword: Angel-SOM-8.2 (See Preface for additional details.)
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