The supplement of an acute angle is an obtuse angle.

PERSPECTIVE ON HISTORY The Development of Geometry One of the first written accounts of geometric knowledge appears in the Rhind papyrus, a collection of documents that date back to more than 1000 years before Christ. In this document, Ahmes an Egyptian scribe describes how north- south and east-west lines were redrawn following the overflow of the Nile River. Astronomy was used to lay out the north-south line. The rest was done by people known as “rope-fasteners.” By tying knots in a rope, it was possible to separate the rope into segments with lengths that were in the ratio 3 to 4 to 5. The knots were fastened at stakes in such a way that a right triangle would be formed. In Figure 1.69, the right angle is formed so that one side of length 4, as shown lies in the north-south line, and the second side of length 3, as shown lies in the east-west line. The principle that was used by the rope-fasteners is known as the Pythagorean Theorem. However, we also know that the ancient Chinese were aware of this relationship. That is, the Pythagorean Theorem was known and applied many centuries before the time of Pythagoras the Greek mathematician for whom the theorem is named. Ahmes describes other facts of geometry that were known to the Egyptians. Perhaps the most impressive of these facts was that their approximation of ␲ was 3.1604. To four decimal places of accuracy, we know today that the correct value of ␲ is 3.1416. Like the Egyptians, the Chinese treated geometry in a very practical way. In their constructions and designs, the Chinese used the rule ruler, the square, the compass, and the level. Unlike the Egyptians and the Chinese, the Greeks formalized and expanded the knowledge base of geometry by pursuing geometry as an intellectual endeavor. According to the Greek scribe Proclus about 50 B . C ., Thales 625–547 B . C . first established deductive proofs for several of the known theorems of geometry. Proclus also notes that it was Euclid 330–275 B . C . who collected, summarized, ordered, and verified the vast quantity of knowledge of geometry in his time. Euclid’s work Elements was the first textbook of geometry. Much of what was found in Elements is the core knowledge of geometry and thus can be found in this textbook as well. N S E W Figure 1.69 PERSPECTIVE ON APPLICATION Patterns In much of the study of mathematics, we seek patterns related to the set of counting numbers N = {1,2,3,4,5, . . .}. Some of these patterns are geometric and are given special names that reflect the configuration of sets of points. For instance, the set of square numbers are shown geometrically in Figure 1.70 and, of course, correspond to the numbers 1, 4, 9, 16, . . . . EXAMPLE 1 Find the fourth number in the pattern of triangular numbers shown in Figure 1.71a. Figure 1.70 Figure 1.71a 1 1 point 3 3 points 6 6 points ? ? points Figure 1.71b Some patterns of geometry lead to principles known as postulates and theorems. One of the principles that we will explore in the next example is based on the total number of diagonals found in a polygon with a given number of sides. A diagonal of a polygon many-sided figure joins two non- consecutive vertices of the polygon together. Of course, joining any two vertices of a triangle will determine a side; thus, a triangle has no diagonals. In Example 2, both the number of sides of the polygon and the number of diagonals are shown. EXAMPLE 2 Find the total number of diagonals for a polygon of 6 sides. 3 sides 0 diagonals 6 sides ? diagonals 5 sides 5 diagonals 4 sides 2 diagonals Figure 1.72a Solution Adding a row of 4 points at the bottom, we have the diagram shown in Figure 1.71b, which contains 10 points. The fourth triangular number is 10. 쮿 쮿 쮿 Solution By drawing all possible diagonals as shown in Figure 1.72b and counting them, we find that there are a total of 9 diagonals Figure 1.72b Certain geometric patterns are used to test students, as in testing for intelligence IQ or on college admissions tests. A simple example might have you predict the next fourth figure in the pattern of squares shown in Figure 1.73a. Figure 1.73a Figure 1.73b We rotate the square once more to obtain the fourth figure as shown in Figure 1.73b. EXAMPLE 3 Midpoints of the sides of a square are used to generate new figures in the sequence shown in Figure 1.74a. Draw the fourth figure. Figure 1.74a Figure 1.74b Solution By continuing to add and join midpoints in the third figure, we form a figure like the one shown in Figure 1.74b. Note that each new figure within the previous figure is also a square Summary A LOOK BACK AT CHAPTER 1 Our goal in this chapter has been to introduce geometry. We discussed the types of reasoning that are used to de- velop geometric relationships. The use of the tools of measurement ruler and protractor was described. We en- countered the four elements of a mathematical system: undefined terms, definitions, postulates, and theorems. The undefined terms were needed to lay the foundation for defining new terms. The postulates were needed to lay the foundation for the theorems we proved here and for the theorems that lie ahead. Constructions presented in this chapter included the bisector of an angle and the per- pendicular to a line at a point on the line. A LOOK AHEAD TO CHAPTER 2 The theorems we will prove in the next chapter are based on a postulate known as the Parallel Postulate. A new method of proof, called indirect proof, will be introduced; it will be used in later chapters. Although many of the the- orems in Chapter 2 deal with parallel lines, several theo- rems in the chapter deal with the angles of a polygon. Symmetry and transformations will be discussed. KEY CONCEPTS 1.1 Statement • Variable • Conjunction • Disjunction • Negation • Implication Conditional • Hypothesis • Conclusion • Intuition • Induction • Deduction • Argument Valid and Invalid • Law of Detachment • Set • Subets • Venn Diagram • Intersection • Union 1.2 Point • Line • Plane • Collinear Points • Vertex • Line Segment • Betweenness of Points • Midpoint • Congruent • Protractor • Parallel Lines • Bisect • Straight Angle • Right Angle • Intersect • Perpendicular • Compass • Constructions • Circle • Arc • Radius 1.3 Mathematical System • Axiom or Postulate • Assumption • Theorem • Ruler Postulate • Distance • Segment-Addition Postulate • Congruent Segments • Midpoint of a Line Segment • Bisector of a Line Segment • Union • Ray • Opposite Rays • Intersection of Two Geometric Figures • Parallel Lines • Plane • Coplanar Points • Space • Parallel, Vertical, Horizontal Planes 1.4 Angle • Sides of an Angle • Vertex of an Angle • Protractor Postulate • Acute, Right, Obtuse, Straight, and Reflex Angles • Angle-Addition Postulate • Adjacent Angles • Congruent Angles • Bisector of an Angle • Complementary Angles • Supplementary Angles • Vertical Angles 1.5 Algebraic Properties • Proof 1.6 Vertical Lines and Horizontal Lines • Perpendicular Lines • Relations • Reflexive, Symmetric, and Transitive Properties of Congurence • Equivalence Relation • Perpendicular Bisector of a Line Segment 1.7 Formal Proof of a Theorem • Converse of a Theorem TABLE 1.9 An Overview of Chapter 1 Line and Line Segment Relationships FIGURE RELATIONSHIP SYMBOLS Parallel lines and segments ᐉ 储 m or 储 ; 储 CD AB Í CD Í AB Intersecting lines 傽 = K Í GH EF Perpendicular lines t shown vertical, v shown horizontal t ⊥ v Congruent line segments ⬵ ; MN = PQ PQ MN Point B between A and C on AC A -B-C; AB + BC = AC Point M the midpoint of PQ ; PM = MQ ; PM = PQ 1 2 MQ ⬵ PM Angle Classification One Angle m A B C D G E K F H v t A B C P M Q M N P Q FIGURE TYPE ANGLE MEASURE Acute angle 0° m⬔1 90° Right angle m ⬔2 = 90° 1 2 continued Angle Relationships Two Angles FIGURE RELATIONSHIP SYMBOLS Congruent angles ⬔1 ⬵ ⬔2; m ⬔1 = m ⬔2 Adjacent angles m ⬔3 + m ⬔4 = m ⬔ABC Bisector of angle bisects ⬔GHJ HK ⬔5 ⬵ ⬔6; m ⬔5 = m ⬔6; m ⬔5 = m ⬔GHJ 1 2 Complementary angles m ⬔7 + m ⬔8 = 90° Supplementary angles m ⬔9 + m ⬔10 = 180° Vertical angles ⬔11 and ⬔12; ⬔13 and ⬔14 ⬔11 ⬵ ⬔12; ⬔13 ⬵ ⬔14 10 9 11 12 13 14 1 2 A B C D 3 4 G H J 6 K 5 7 8 Angle Classification One Angle FIGURE TYPE ANGLE MEASURE Obtuse angle 90° m⬔3 180° Straight angle m ⬔4 = 180° Reflex angle 180° m⬔5 360° 4 5 TABLE 1.9 continued 3

1. Name the four components of a mathematical system. 2. Name three types of reasoning.

3. Name the four characteristics of a good definition.

In Review Exercises 4 to 6, name the type of reasoning illustrated.

4. While watching the pitcher warm up, Phillip thinks, “I’ll

be able to hit against him.”

5. Laura is away at camp. On the first day, her mother brings

her additional clothing. On the second day, her mother brings her another pair of shoes. On the third day, her mother brings her cookies. Laura concludes that her mother will bring her something on the fourth day.

6. Sarah knows the rule “A number not 0 divided by itself

equals 1.” The teacher asks Sarah, “What is 5 divided by 5?” Sarah says, “The answer is 1.” In Review Exercises 7 and 8, state the hypothesis and conclusion for each statement.

7. If the diagonals of a trapezoid are equal in length, then the

trapezoid is isosceles.

8. The diagonals of a parallelogram are congruent if the

parallelogram is a rectangle. In Review Exercises 9 to 11, draw a valid conclusion where possible.

9. 1. If a person has a good job, then that person has a

college degree. 2. Billy Fuller has a college degree. C. ‹ ?

10. 1. If a person has a good job, then that person has a

college degree. 2. Jody Smithers has a good job. C. ‹ ?

11. 1. If the measure of an angle is 90°, then that angle is a

right angle. 2. Angle A has a measure of 90°. C. ‹ ?

12. A, B, and C are three points on a line. AC

= 8, BC = 4, and AB = 12. Which point must be between the other two points?

13. Use three letters to name the angle shown. Also use one

letter to name the same angle. Decide whether the angle measure is less than 90°, equal to 90°, or greater than 90°.

14. Figure MNPQ is a rhombus. Draw diagonals and

of the rhombus. How do and appear to be related? QN MP QN MP Chapter 1 REVIEW EXERCISES S T R In Review Exercises 15 to 17, sketch and label the figures described.

15. Points A, B, C, and D are coplanar. A, B, and C are the

only three of these points that are collinear.

16. Line intersects plane X at point P.

17. Plane M contains intersecting lines j and k. 18. On the basis of appearance, what type of angle is shown? Q P M N 1 a 2 b 4 a 5 b 19. On the basis of appearance, what type of angle is shown? Exercises 20, 21 Exercises 22, 23 C B D A

20. Given:

bisects ⬔ABC m ⬔ABD = 2x + 15 m ⬔DBC = 3x - 2 Find: m ⬔ABC

21. Given:

m ⬔ABD = 2x +

5 m

⬔DBC = 3x - 4 m ⬔ABC = 86° Find: m ⬔DBC

22. Given: AM

= 3x - 1 MB = 4x - 5 M is the midpoint of Find: AB

23. Given: AM

= 4x - 4 MB = 5x + 2 AB = 25 Find: MB

24. Given: D

is the midpoint of ⬵ CD = 2x + 5 BC = x + 28 Find: AC BC AC AC AB BD A B M C B D A

25. Given:

m ⬔3 = 7x - 21 m ⬔4 = 3x + 7 Find: m ⬔FMH

26. Given:

m ⬔FMH = 4x + 1 m ⬔4 = x + 4 Find: m ⬔4

27. In the figure, find:

a 傽 b 傼 c ⬔KMJ 傽 ⬔JMH d 傼

28. Given:

⬔EFG is a right angle m ⬔HFG = 2x - 6 m ⬔EFH = Find: m ⬔EFH

29. Two angles are supplementary. One angle is 40° more

than four times the other. Find the measures of the two angles.

30. a Write an expression for the perimeter of the triangle

shown. HINT: Add the lengths of the sides. b If the perimeter is 32 centimeters, find the value of x. c Find the length of each side of the triangle. 3 m ⬔HFG MH MK MH MJ Í FJ Í KH

38. Fill in the missing statements or reasons.

Given: ⬔1 ⬵ ⬔P ⬔4 ⬵ ⬔P bisects ⬔RVO Prove: ⬔TVP ⬵ ⬔MVP VP K H 4 3 M J F Exercises 25–27 x + 7 3x – 2 2x + 3 1 2 3

31. The sum of the measures of all three angles of the tri-

angle in Review Exercise 30 is 180°. If the sum of the measures of angles 1 and 2 is more than 130°, what can you conclude about the measure of angle 3?

32. Susan wants to have a 4-ft board with some pegs on it.

She wants to leave 6 in. on each end and 4 in. between pegs. How many pegs will fit on the board? HINT: If n represents the number of pegs, then n - 1 represents the number of equal spaces. State whether the sentences in Review Exercises 33 to 37 are always true A, sometimes true S, or never true N.

33. If AM

= MB , then A, M, and B are collinear. 34. If two angles are congruent, then they are right angles. 35. The bisectors of vertical angles are opposite rays.

36. Complementary angles are congruent. 37. The supplement of an obtuse angle is another obtuse

angle.

3 4

1 2 T M O P R V PROOF Statements Reasons 1. ⬔1 ⬵ ⬔P 1. Given 2. ? 2. Given 1, 2 3. ? 3. Transitive Prop. of ⬵ 3 4. m ⬔1 = m ⬔4 4. ? 5. bisects ⬔RVO 5. ? 6. ? 6. If a ray bisects an ⬔, it forms two ⬔s of equal measure 4, 6 7. ? 7. Addition Prop. of Equality 8. m ⬔1 + m ⬔2 = 8. ? m ⬔TVP; m ⬔4 + m ⬔3 = m ⬔MVP 7, 8 9. m ⬔TVP = 9. ? m ⬔MVP 10. ? 10. If two ⬔s are = in measure, then they are ⬵ Write two-column proofs for Review Exercises 39 to 46. VP K J F H G Exercises 39–41

39. Given: ⊥

⬔JHF is a right ⬔ Prove: ⬔KFH ⬵ ⬔JHF FH KF G F H E