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7.4 Lines and Angles 7.4 OBJECTIVES 1. 2. 3. 4. NOTE “Geo” means earth, just as it does in the words “geography” and “geology.” Distinguish between lines and line segments Determine when lines are perpendicular or parallel Determine whether an angle is right, acute, or obtuse Use a protractor to measure an angle After counting, there was geometry. Once the Egyptians and Babylonians had mastered the counting of their animals, they became interested in measuring their land. This is the foundation of geometry. Literally translated, “geometry” means earth measurement. Many of the topics we consider in geometry (topics such as angles, perimeter, and area) were first studied as part of surveying. As is usually the case, we start the study of a new topic by learning some vocabulary. Most of the terms we will discuss will be familiar to you. It is important that you understand what we mean when we use these words in the context of geometry. We begin with the word “point.” A point is a location; it has no size and covers no area. If we string points together forever, we create a line. In our studies we will consider only straight lines. We use arrowheads to indicate that a line goes on forever. A piece of a line that has two endpoints is called a line segment. Example 1 Recognizing Lines and Line Segments NOTE The capital letters are Label each of the following as a line or a line segment. labels for points. C A E B F D (c) (b) (a) Both a and c continue forever in both directions. They are lines. Part b has two endpoints. It is a line segment. © 2001 McGraw-Hill Companies CHECK YOURSELF 1 Label each of the following as a line or a line segment. F B C A (1) E (2) (3) 575 CHAPTER 7 GEOMETRY AND MEASURE Definitions: Angle An angle is a geometric figure consisting of two line segments that share a common endpoint. A O B OA and OB are line segments. O is the vertex of the angle. Surveyors use an instrument called a transit. A transit allows surveyors to measure angles so that, from a mathematical description, they can determine exactly where a property line is. Definitions: Perpendicular Lines When two lines cross ( or intersect) they form four angles. If the lines intersect such that four equal angles are formed, we say that the two lines are perpendicular. At most highway intersections, the two roads are perpendicular. Definitions: Parallel Lines If two lines are drawn so that they never intersect (even if we extend the lines forever), we say that the two lines are parallel. Parallel parking gets its name from the fact that the parking spot is parallel to the traffic lane. © 2001 McGraw-Hill Companies 576 LINES AND ANGLES SECTION 7.4 577 Example 2 Recognizing Parallel and Perpendicular Lines Label each pair of lines as parallel, perpendicular, or neither. (a) (c) (b) Although we don’t see the lines in part a intersecting, if they were extended as the arrowheads indicate, they would. The lines of part b are perpendicular because the four angles formed are equal. Only the lines in part c are parallel. CHECK YOURSELF 2 Label each pair of lines as parallel, perpendicular, or neither. (1) NOTE You may recall seeing this small square in Chapter 4. There we used it to show the altitude (height) of a triangle. (3) (2) We call the angle formed by two perpendicular lines or line segments a right angle. We designate a right angle by forming a small square. A B O We can refer to a specific angle by naming three points. The middle point is the vertex of the angle. Example 3 © 2001 McGraw-Hill Companies Naming an Angle Name the highlighted angle. B A C O D NOTE We could also call this angle BOC. The vertex of the angle is O, and the angle begins at C and ends at B, so we would name the angle COB. 578 CHAPTER 7 GEOMETRY AND MEASURE CHECK YOURSELF 3 Name the highlighted angle. B C O A D One way to measure an angle is to use a unit that we call a degree. There are 360 degrees (we write this as 360°) in a complete circle. Note in the picture on the left that there are four right angles in a circle. If we divide 360° by 4, we find that each right angle must measure 90°. Here are some other angles with their measurements. 120 60 30 180 An acute angle measures between 0° and 90°. An obtuse angle measures between 90° and 180°. A straight angle measures 180°. Example 4 Labeling Types of Angles Label each of the following angles as an acute, obtuse, right, or straight angle. (a) (b) (c) (d) Part a is obtuse (the angle is more than 90°). Part b is a right angle (designated by the small square). Part c is an acute angle (it is less than 90°), and part d is a straight angle. Label each angle as an acute, an obtuse, a right, or a straight angle. (1) (2) (3) (4) © 2001 McGraw-Hill Companies CHECK YOURSELF 4 LINES AND ANGLES SECTION 7.4 579 When assigning a measurement to an angle, we usually use a tool called a protractor. NOTE Your protractor may show the degree measures in both directions. 100 110 80 70 90 80 70 90 100 1 60 10 12 0 50 13 0 0 10 20 0 180 30 160 17 0 15 40 0 14 180 170 160 0 10 15 0 20 14 30 0 40 0 12 0 60 13 50 Place the protractor so that the vertex of the angle is here. We read the protractor by placing one line segment of the angle at 0°. We then read the number that the other line segment passes through. This number represents the degree measurement of the angle. The point at the center of the protractor, the endpoint of the two line segments, is the vertex of the angle. Example 5 Measuring an Angle Use the protractor to estimate the measurement for each angle. F B D A O C E O O The measure of AOB is 45°. The measure of COD is 150°. The measure of EOF is between 50° and 55°. We could estimate that it is a 52° angle. CHECK YOURSELF 5 Use a protractor to estimate the measurement for each angle. D © 2001 McGraw-Hill Companies B A O (2) F E O (3) C O (1) 580 CHAPTER 7 GEOMETRY AND MEASURE If we wish to refer to the degree measure of ABC, we use mABC. Example 6 Measuring an Angle Find mAOB. B C A O D NOTE mAOB 20° is read Using the protractor, we find mAOB 20°. “the measure of angle AOB is 20 degrees.” CHECK YOURSELF 6 Find mAOC. C B D A O E F CHECK YOURSELF ANSWERS (1) Line segment; (2) line segment; (3) line (1) Parallel; (2) neither; (3) perpendicular (1) Right; (2) straight; (3) acute; (4) obtuse (1) 120°; (2) 80°; (3) 160° 6. 135° 3. BOA or AOB © 2001 McGraw-Hill Companies 1. 2. 4. 5. Name Exercises 7.4 Section 1. Draw line segment AB. # A 2. Draw line EF. # B # E 3. Draw line AC. # A Date # F ANSWERS 1. 4. Draw line segment BC. # C # B # C 2. 3. Identify each object as a line or line segment. 4. 5. 6. 7. P D 5. U 6. O C V 7. 8. 8. 9. 10. A X K 9. 10. L W 11. 12. B 13. 11. 12. H E 14. 15. F G 16. 17. Label exercises 13 to 18 as true or false. 18. 13. There are exactly two different line segments that can be drawn through two points. © 2001 McGraw-Hill Companies 14. There are exactly two different lines that can be drawn through two points. 15. Two opposite sides of a square are parallel line segments. 16. Two adjacent sides of a square are perpendicular line segments. 17. ABC will always have the same measure as CAB. 18. Two acute angles have the same measure. 581 ANSWERS 19. 19. Are the following two lines parallel, perpendicular, or neither? 20. 21. 22. 23. 20. Are the following two lines parallel, perpendicular, or neither? 24. 25. 26. 27. 28. 29. Give an appropriate name for each indicated angle. 21. 30. 22. Q P 23. U M V R 31. 32. N S O L T 24. 25. A F 26. G X Y Z H E B W C 27. 28. S J K R L I V N U M For each angle described, give its measure in degrees. One revolution is a full circle. Sketch the angle. 582 29. A represents 1 of a revolution 6 30. B represents 1 of a revolution 3 31. C represents 7 of a revolution 12 32. D represents 11 of a revolution 12 © 2001 McGraw-Hill Companies T ANSWERS 33. Measure each angle with a protractor. Identify the angle as acute, right, obtuse, or straight. 33. 34. A 34. E 35. D 36. 37. B O F 38. 39. 35. 36. O P 40. D C R Q 37. 38. F G O E D F In the figure, two parallel lines are intersected by a third line, forming eight angles. Draw lines like these on your paper. 1 © 2001 McGraw-Hill Companies 3 5 7 2 4 6 8 39. Use your protractor to measure 2 and 6. What do you notice? 40. Use your protractor to measure 3 and 6. What do you notice? 583 ANSWERS 41. Draw any triangle using a ruler. With your protractor, carefully measure the three 41. interior angles, and find their sum. Do this again with two more triangles of different shapes. What do you notice about the sums of the angles? Make a conjecture about the sum of the angles of any triangle. 42. 42. A quadrilateral is a four-sided polygon. Draw any quadrilateral, and measure the 43. four interior angles with a protractor. Record these, and find their sum. Make a conjecture concerning the sum of the interior angles of any quadrilateral. Test your conjecture on another quadrilateral. 44. 43. A pentagon is a five-sided polygon. Draw any pentagon, and measure the five interior angles with a protractor. Record these, and find their sum. Make a conjecture concerning the sum of the interior angles of any pentagon. Test your conjecture on another pentagon. 44. A hexagon is a six-sided polygon. Draw any hexagon, and measure the six interior angles with a protractor. Record these, and find their sum. Make a conjecture concerning the sum of the interior angles of any hexagon. Test your conjecture on another hexagon. Answers 1. A B 3. A C 5. Line © 2001 McGraw-Hill Companies 7. Line segment 9. Line segment 11. Line 13. False 15. True 17. False 19. Parallel 21. POQ 23. MNL 25. FEG 27. SVT 29. 60° 31. 210° 33. 135°; obtuse 35. 90°; right 37. 30°; acute 39. 40°; 40° 41. 43. 584