Using the number line provided, name the point that

36. Consider noncoplanar points A, B, C, and D. Using three

points at a time such as A, B, and C, how many planes are determined by these points?

37. Line is parallel to plane P that is, it will not intersect P

even if extended. Line m intersects line . What can you conclude about m and P?

28. Consider the figure for Exercise 27. Given that B is the

midpoint of and C is the midpoint of , what can you conclude about the lengths of a and ? c and ? b and ? In Exercises 29 to 32, use only a compass and a straightedge to complete each construction.

29. Given: and AB ⬎ CD

Construct: on line so that MN = AB + CD MN CD AB BD AC CD AC CD AB BD AC

30. Given: and AB ⬎ CD

Construct: so that EF = AB - CD

31. Given: as shown in the figure

Construct: on line n so that PQ = 3AB PQ AB EF CD AB A B C D Exercises 29, 30

32. Given: as shown in the figure

Construct: on line n so that AB

33. Can you use the construction for the midpoint of a

segment to divide a line segment into a three congruent parts? c six congruent parts? b four congruent parts? d eight congruent parts? 34. Generalize your findings in Exercise 33. 35. Consider points A, B, C, and D, no three of which are collinear. Using two points at a time such as A and B, how many lines are determined by these points? TV = 1 2 TV AB n A B Exercises 31, 32 P m

38. and

are said to be skew lines because they neither intersect nor are parallel. How many planes are determined by a parallel lines AB and DC? b intersecting lines AB and BC? c skew lines AB and EF? d lines AB, BC, and DC? e points A, B, and F? f points A, C, and H? g points A, C, F, and H? Í EF Í AB A B D C E F G H

39. Let AB

= a and BC = b . Point M is the midpoint of . If AB, find the length of in terms of a and b. NM AN = 2 3 BC C A B M N This section introduces you to the language of angles. Recall from Sections 1.1 and 1.3 that the word union means that two sets or figures are joined. Angle: Sides of Angle, Vertex of 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 Angles and Their Relationships 1.4 KEY CONCEPTS An angle is the union of two rays that share a common endpoint. DEFINITION In Figure 1.46, the angle is symbolized by ⬔ABC or ⬔CBA. The rays BA and BC are known as the sides of the angle. B, the common endpoint of these rays, is known as the vertex of the angle. When three letters are used to name an angle, the vertex is al- ways named in the middle. Recall that a single letter or numeral may be used to name the angle. The angle in Figure 1.46 may be described as ⬔B the vertex of the angle or as ⬔1. In set notation, ⬔B = . BA ´ BC 1 B C A Figure 1.46 B C E A F D Figure 1.47 C B D A Figure 1.48 The measure of an angle is a unique positive number. POSTULATE 8 왘 Protractor Postulate NOTE: In Chapters 1 to 10, the measures of most angles will be between 0° and 180°, including 180°. Angles with measures between 180° and 360° are introduced in this section; these angles are not used often in our study of geometry. TYPES OF ANGLES An angle whose measure is less than 90° is an acute angle. If the angle’s measure is ex- actly 90°, the angle is a right angle. If the angle’s measure is between 90° and 180°, the angle is obtuse. An angle whose measure is exactly 180° is a straight angle; alterna- tively, a straight angle is one whose sides form opposite rays a straight line. A reflex angle is one whose measure is between 180° and 360°. See Table 1.4 on page 32. In Figure 1.47, ⬔ABC contains the noncollinear points A, B, and C. These three points, in turn, determine a plane. The plane containing ⬔ABC is separated into three subsets by the angle: Points like D are said to be in the interior of ⬔ABC. Points like E are said to be on ⬔ABC. Points like F are said to be in the exterior of ⬔ABC. With this description, it is possible to state the counterpart of the Segment-Addition Postulate Consider Figure 1.48 as you read Postulate 9. If a point D lies in the interior of an angle ABC, then m ⬔ABD + m ⬔DBC = m ⬔ABC. POSTULATE 9 왘 Angle-Addition Postulate