In the figure, a b c and B is

first unit square; in Figure 4.41b, the “wrap-around” is indicated by 3 shaded squares. Now for the second square sides of length 2, we form the next square by wrapping 5 unit squares around this square; see Figure 4.41c. The next figure in the sequence of squares illustrates that In the “wrap-around,” we emphasize that the next number in the sum is an odd number. The “wrap-around” approach adds , or 7 unit squares in Figure 4.42. When building each sequential square, we always add an odd number of unit squares as in Figure 4.42. 2 3 + 1 1 + 3 + 5 + 7 = 4 2 , or 16 Figure 4.42 PROBLEM Use the following principle to answer each question: Where n is a counting number, the sum of the first n positive odd counting numbers is . a Find the sum of the first five positive odd integers; that is, find . b Find the sum of the first six positive odd integers. c How many positive odd integers were added to obtain the sum 81? Solutions a , or 25 b , or 36 c 9, because 9 2 = 81 6 2 5 2 1 + 3 + 5 + 7 + 9 n 2 쮿 Summary A LOOK BACK AT CHAPTER 4 The goal of this chapter has been to develop the proper- ties of quadrilaterals, including special types of quadrilat- erals such as the parallelogram, rectangle, and trapezoid. Table 4.1 on page 213 summarizes the properties of quadrilaterals. A LOOK AHEAD TO CHAPTER 5 In the next chapter, similarity will be defined for all poly- gons, with an emphasis on triangles. The Pythagorean Theorem, which we applied in Chapter 4, will be proved in Chapter 5. Special right triangles will be discussed. KEY CONCEPTS

4.1

Quadrilateral • Skew Quadrilateral • Parallelogram • Diagonals of a Parallelogram • Altitudes of a Parallelogram 4.2 Quadrilaterals That Are Parallelograms • Rectangle • Kite 4.3 Rectangle • Square • Rhombus • Pythagorean Theorem 4.4 Trapezoid Bases, Legs, Base Angles, Median • Isosceles Trapezoid Properties of Quadrilaterals PARALLELO- GRAM RECTANGLE RHOMBUS SQUARE KITE TRAPEZOID ISOSCELES TRAPEZOID Congruent sides Both pairs of opposite sides Both pairs of opposite sides All four sides All four sides Both pairs of adjacent sides Possible; also see isosceles trapezoid Pair of legs Parallel sides Both pairs of opposite sides Both pairs of opposite sides Both pairs of opposite sides Both pairs of opposite sides Generally none Pair of bases Pair of bases Perpendicular sides If the parallelogram is a rectangle or square Consecutive pairs If rhombus is a square Consecutive pairs Possible Possible Generally none Congruent angles Both pairs of opposite angles All four angles Both pairs of opposite angles All four angles One pair of opposite angles Possible; also see isosceles trapezoid Each pair of base angles Supplemen- tary angles All pairs of consecutive angles Any two angles All pairs of consecutive angles Any two angles Possibly two pairs Each pair of leg angles Each pair of leg angles Diagonal relationships Bisect each other Congruent; bisect each other Perpendicular; bisect each other and interior angles Congruent; perpendi- cular; bisect each other and interior angles Perpendi- cular; one bisects other and two interior angles Intersect Congruent TABLE 4.1 An Overview of Chapter 4