Lines and Angles. Example 1 Recognizing Lines and Line Segments. Label each of the following as a line or a line segment. A E.

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1 7.4 Lines and ngles 7.4 JTIVS 1. istinguish between lines and line segments 2. etermine when lines are perpendicular or parallel 3. etermine whether an angle is right, acute, or obtuse 4. Use a protractor to measure an angle NT Geo means earth, just as it does in the words geography and geology. fter counting, there was geometry. nce the gyptians and abylonians 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. s 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. 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. piece of a line that has two endpoints is called a line segment. xample 1 Recognizing Lines and Line Segments NT The capital letters are labels for points. Label each of the following as a line or a line segment. (a) (b) (c) oth a and c continue forever in both directions. They are lines. Part b has two endpoints. It is a line segment. HK YURSL 1 Label each of the following as a line or a line segment McGraw-Hill ompanies (1) (2) (3) 575

2 576 HPTR 7 GMTRY N MSUR efinitions: ngle n angle is a geometric figure consisting of two line segments that share a common endpoint. and are line segments. is the vertex of the angle. Surveyors use an instrument called a transit. transit allows surveyors to measure angles so that, from a mathematical description, they can determine exactly where a property line is. efinitions: 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. t most highway intersections, the two roads are perpendicular. efinitions: 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 McGraw-Hill ompanies

3 LINS N NGLS STIN xample 2 Recognizing Parallel and Perpendicular Lines Label each pair of lines as parallel, perpendicular, or neither. (a) (b) (c) lthough 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. nly the lines in part c are parallel. HK YURSL 2 Label each pair of lines as parallel, perpendicular, or neither. (1) (2) (3) NT You may recall seeing this small square in hapter 4. There we used it to show the altitude (height) of a triangle. 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. We can refer to a specific angle by naming three points. The middle point is the vertex of the angle. xample 3 Naming an ngle 2001 McGraw-Hill ompanies NT We could also call this angle. Name the highlighted angle. The vertex of the angle is, and the angle begins at and ends at, so we would name the angle.

4 578 HPTR 7 GMTRY N MSUR HK YURSL 3 Name the highlighted angle. ne 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 n acute angle measures between 0 and 90. n obtuse angle measures between 90 and 180. straight angle measures 180. xample 4 Labeling Types of ngles 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. HK YURSL 4 (1) Label each angle as an acute, an obtuse, a right, or a straight angle. (2) (3) (4) 2001 McGraw-Hill ompanies

5 LINS N NGLS STIN NT Your protractor may show the degree measures in both directions. When assigning a measurement to an angle, we usually use a tool called a protractor 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. xample 5 Measuring an ngle Use the protractor to estimate the measurement for each angle. The measure of is 45. The measure of is 150. The measure of is between 50 and 55. We could estimate that it is a 52 angle. HK YURSL 5 Use a protractor to estimate the measurement for each angle McGraw-Hill ompanies (1) (3) (2)

6 580 HPTR 7 GMTRY N MSUR If we wish to refer to the degree measure of, we use m. xample 6 Measuring an ngle ind m. NT m 20 is read the measure of angle is 20 degrees. Using the protractor, we find m 20. HK YURSL 6 ind m. HK YURSL NSWRS 1. (1) Line segment; (2) line segment; (3) line 2. (1) Parallel; (2) neither; (3) perpendicular 3. or 4. (1) Right; (2) straight; (3) acute; (4) obtuse 5. (1) 120 ; (2) 80 ; (3) McGraw-Hill ompanies

7 Name 7.4 xercises Section ate 1. raw line segment. 2. raw line. # # # # 3. raw line. 4. raw line segment. # # # # Identify each object as a line or line segment P U NSWRS V K X L W H G Label exercises 13 to 18 as true or false. 13. There are exactly two different line segments that can be drawn through two points There are exactly two different lines that can be drawn through two points McGraw-Hill ompanies 15. Two opposite sides of a square are parallel line segments. 16. Two adjacent sides of a square are perpendicular line segments. 17. will always have the same measure as. 18. Two acute angles have the same measure. 581

8 NSWRS re the following two lines parallel, perpendicular, or neither? re the following two lines parallel, perpendicular, or neither? Give an appropriate name for each indicated angle P Q U V R M N 32. T S L G 26. X Y Z H W 27. S 28. J K R T I L V U N M or each angle described, give its measure in degrees. ne revolution is a full circle. Sketch the angle represents of a revolution 30. represents of a revolution represents of a revolution 32. represents of a revolution McGraw-Hill ompanies 582

9 NSWRS Measure each angle with a protractor. Identify the angle as acute, right, obtuse, or straight P Q R G In the figure, two parallel lines are intersected by a third line, forming eight angles. raw lines like these on your paper McGraw-Hill ompanies 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

10 NSWRS raw any triangle using a ruler. With your protractor, carefully measure the three interior angles, and find their sum. o 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 quadrilateral is a four-sided polygon. raw any quadrilateral, and measure the 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. 43. pentagon is a five-sided polygon. raw 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. hexagon is a six-sided polygon. raw 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. nswers Line 7. Line segment 9. Line segment 11. Line 13. alse 15. True 17. alse 19. Parallel 21. PQ 23. MNL 25. G 27. SVT ; obtuse ; right ; acute ; McGraw-Hill ompanies 584

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