In the drawing, m

For Exercises 49 and 50, use the following information. Relative to its point of departure or some other point of reference, the angle that is used to locate the position of a ship or airplane is called its bearing. The bearing may also be used to describe the direction in which the airplane or ship is moving. By using an angle between 0° and 90°, a bearing is measured from the North-South line toward the East or West. In the diagram, airplane A which is 250 miles from Chicago’s O’Hare airport’s control tower has a bearing of S 53° W.

49. Find the bearing of airplane B relative to the control

tower.

50. Find the bearing of airplane C relative to the control

tower. 53 250 mi 325 mi 300 mi 24 22 control tower W E S N B C A Exercises 49, 50 A MATHEMATICAL SYSTEM Like algebra, the branch of mathematics called geometry is a mathematical system. Each system has its own vocabulary and properties. In the formal study of a mathemat- ical system, we begin with undefined terms. Building on this foundation, we can then define additional terms. Once the terminology is sufficiently developed, certain prop- erties characteristics of the system become apparent. These properties are known as axioms or postulates of the system; more generally, such statements are called as- sumptions. Once we have developed a vocabulary and accepted certain postulates, many principles follow logically when we apply deductive methods. These statements can be proved and are called theorems. The following box summarizes the components of a mathematical system sometimes called a logical system or deductive system. Mathematical System Axiom or Postulate Theorem Ruler Postulate Distance Segment-Addition Postulate Midpoint of a Line Segment Ray Opposite Rays Intersection of Two Geometric Figures Plane Coplanar Points Space Early Definitions and Postulates 1.3 KEY CONCEPTS FOUR PARTS OF A MATHEMATICAL SYSTEM 1. Undefined terms

2. Defined terms

f vocabulary

3. Axioms or postulates 4. Theorems

f principles line; plane In many textbooks, it is common to use the phrase “if and only if ” in expressing the definition of a term. For instance, we could define congruent angles by saying that two angles are congruent if and only if these angles have equal measures. The “if and only if ” statement has the following dual meaning: “If two angles are congruent, then they have equal measures.” “If two angles have equal measures, then they are congruent.” When represented by a Venn Diagram, this definition would relate set C = {con- gruent angles} to set E = {angles with equal measures} as shown in Figure 1.30. The sets C and E are identical and are known as equivalent sets. Once undefined terms have been described, they become the building blocks for other terminology. In this textbook, primary terms are defined within boxes, whereas related terms are often boldfaced and defined within statements. Consider the follow- ing definition see Figure 1.31. CHARACTERISTICS OF A GOOD DEFINITION Terms such as point, line, and plane are classified as undefined because they do not fit into any set or category that has been previously determined. Terms that are defined, however, should be described precisely. But what is a good definition? A good defini- tion is like a mathematical equation written using words. A good definition must possess four characteristics. We illustrate this with a term that we will redefine at a later time. An isosceles triangle is a triangle that has two congruent sides. DEFINITION Discover Although we cannot actually define line and plane, we can compare them in the following analogy. Please complete: A ___ ? is to straight as a ___ ? is to flat. ANSWERS In the definition, notice that: 1 The term being defined—isosceles triangle—is named. 2 The term being defined is placed into a larger category a type of triangle. 3 The distinguishing quality that two sides of the triangle are congruent is included. 4 The reversibility of the definition is illustrated by these statements: “If a triangle is isosceles, then it has two congruent sides.” “If a triangle has two congruent sides, then it is an isosceles triangle.” IN SUMMARY, A GOOD DEFINITION WILL POSSESS THESE QUALITIES 1. It names the term being defined. 2. It places the term into a set or category. 3. It distinguishes the defined term from other terms without providing unnecessary facts.

4. It is reversible.

C E Figure 1.30 B A Figure 1.31 A line segment is the part of a line that consists of two points, known as endpoints, and all points between them. DEFINITION Exs. 1–4 Considering this definition, we see that 1. The term being defined, line segment, is clearly present in the definition. 2. A line segment is defined as part of a line a category.

3. The definition distinguishes the line segment as a specific part of a line.