Reading to Learn Mathematics

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1 NME TE ERIO 1 Reading to Learn Mathematics Vocabulary uilder This is an alphabetical list of the key vocabulary terms you will learn in hapter 1. s you study the chapter, complete each term s definition or description. Remember to add the page number where you found the term. dd these pages to your Geometry Study Notebook to review vocabulary at the end of the chapter. acute angle Vocabulary Term Found on age efinition/escription/example Vocabulary uilder adjacent angles uh JY suhnt angle angle bisector collinear koh LIN ee uhr complementary angles congruent kuhn GROO uhnt coplanar koh LY nuhr line segment linear pair (continued on the next page) Glencoe/McGraw-Hill vii Glencoe Geometry

angle perimeter perpendicular lines polygon HL ee gahn ray right angle segment

2 NME TE ERIO 1 Reading to Learn Mathematics Vocabulary uilder (continued) midpoint Vocabulary Term Found on age efinition/escription/example obtuse angle perimeter perpendicular lines polygon HL ee gahn ray right angle segment bisector supplementary angles vertical angles Glencoe/McGraw-Hill viii Glencoe Geometry

3 NME TE ERIO 1-1 Study Guide and Intervention oints, Lines, and lanes Name oints, Lines, and lanes In geometry, a point is a location, a line contains points, and a plane is a flat surface that contains points and lines. If points are on the same line, they are collinear. If points on are the same plane, they are coplanar. Example a. a line containing point Use the figure to name each of the following. The line can be named as. lso, any two of the three points on the line can be used to name it.,, or b. a plane containing point The plane can be named as plane N or can be named using three noncollinear points in the plane, such as plane, plane, and so on. N Lesson 1-1 Exercises Refer to the figure. 1. Name a line that contains point. 2. What is another name for line m? m E 3. Name a point not on. 4. Name the intersection of and. 5. Name a point not on line or line m. raw and label a plane Q for each relationship. 6. is in plane Q. 7. ST intersects at. Q T S X Y 8. oint X is collinear with points and. 9. oint Y is not collinear with points T and. 10. Line contains points X and Y. Glencoe/McGraw-Hill 1 Glencoe Geometry

4 NME TE ERIO 1-1 Study Guide and Intervention (continued) oints, Lines, and lanes oints, Lines, and lanes in Space Space is a boundless, three-dimensional set of all points. It contains lines and planes. Example a. How many planes appear in the figure? O N There are three planes: plane N, plane O, and plane. b. re points,, and coplanar? Yes. They are contained in plane O. Exercises Refer to the figure. 1. Name a line that is not contained in plane N. 2. Name a plane that contains point. N E 3. Name three collinear points. Refer to the figure. 4. How many planes are shown in the figure? 5. re points, E, G, and H coplanar? Explain. G F J H I E 6. Name a point coplanar with,, and E. raw and label a figure for each relationship. 7. lanes M andn intersect in HJ. t M s 8. Line r is in plane N, line s is in plane M, and lines r and s intersect at point J. N H J r 9. Line t contains point H and line t does not lie in plane M or plane N. Glencoe/McGraw-Hill 2 Glencoe Geometry

5 1-1 NME TE ERIO Skills ractice Refer to the figure. oints, Lines, and lanes 1. Name a line that contains point. 2. Name a point contained in line n. p n G 3. What is another name for line p? 4. Name the plane containing lines n and p. Lesson 1-1 raw and label a figure for each relationship. 5. oint K lies on RT. 6. lane J contains line s. K R T J s 7. Y lies in plane and contains point, but does not contain point H. in plane U. Y H 8. Lines q and f intersect at point Z U q Z f Refer to the figure. 9. How many planes are shown in the figure? 10. How many of the planes contain points F and E? F E W 11. Name four points that are coplanar. 12. re points,, and coplanar? Explain. Glencoe/McGraw-Hill 3 Glencoe Geometry

6 1-1 NME TE ERIO ractice Refer to the figure. oints, Lines, and lanes 1. Name a line that contains points T and. 2. Name a line that intersects the plane containing points Q, N, and. S R j M T Q N g h 3. Name the plane that contains TN and QR. raw and label a figure for each relationship. 4. K and G intersect at point M in plane T. T M K G 5. line contains L( 4, 4) and M(2, 3). Line q is in the same coordinate plane but does not intersect LM. Line q contains point N. y M q O x L N Refer to the figure. T Q 6. How many planes are shown in the figure? 7. Name three collinear points. W S X R 8. re points N, R, S, and W coplanar? Explain. M N VISULIZTION Name the geometric term(s) modeled by each object tip of pin 11. STO strings 12. a car antenna 13. a library card Glencoe/McGraw-Hill 4 Glencoe Geometry

7 1-1 NME TE ERIO Reading to Learn Mathematics oints, Lines, and lanes re-ctivity Why do chairs sometimes wobble? Reading the Lesson 1. omplete each sentence. Read the introduction to Lesson 1-1 at the top of page 6 in your textbook. Find three pencils of different lengths and hold them upright on your desk so that the three pencil points do not lie along a single line. an you place a flat sheet of paper or cardboard so that it touches all three pencil points? How many ways can you do this if you keep the pencil points in the same position? How will your answer change if there are four pencil points? Lesson 1-1 a. oints that lie on the same lie are called points. b. oints that do not lie in the same plane are called points. c. There is exactly one through any two points. d. There is exactly one through any three noncollinear points. 2. Refer to the figure at the right. Indicate whether each statement is true or false. a. oints,, and are collinear. b. The intersection of plane and line m is point. U c. Line and line m do not intersect. m d. oints,,and can be used to name plane U. e. Line lies in plane. 3. omplete the figure at the right to show the following relationship: Lines, m, and n are coplanar and lie in plane Q. Lines and m intersect at point. Line n intersects line m at R, but does not intersect line. Q R m n Helping You Remember 4. Recall or look in a dictionary to find the meaning of the prefix co-.what does this prefix mean? How can it help you remember the meaning of collinear? Glencoe/McGraw-Hill 5 Glencoe Geometry

8 1-1 NME TE ERIO Enrichment oints and Lines on a Matrix matrix is a rectangular array of rows and columns. oints and lines on a matrix are not defined in the same way as in Euclidean geometry. point on a matrix is a dot, which can be small or large. line on a matrix is a path of dots that line up. etween two points on a line there may or may not be other points. Three examples of lines are shown at the upper right. The broad line can be thought of as a single line or as two narrow lines side by side. ot-matrix printers for computers used dots to form characters. The dots are often called pixels. The matrix at the right shows how a dot-matrix printer might print the letter. raw points on each matrix to create the given figures. 1. raw two intersecting lines that have 2. raw two lines that cross but have four points in common. no common points. 3. Make the number 0 (zero) so that it 4. Make the capital letter O so that it extends to the top and bottom sides extends to each side of the matrix. of the matrix. 5. Using separate grid paper, make dot designs for several other letters. Which were the easiest and which were the most difficult? Glencoe/McGraw-Hill 6 Glencoe Geometry

9 1-2 NME TE ERIO Study Guide and Intervention Linear Measure and recision Measure Line Segments part of a line between two endpoints is called a line segment. The lengths of M N and R S are written as MN and RS.When you measure a segment, the precision of the measurement is half of the smallest unit on the ruler. Example 1 Example 2 Find the length of M N. M N R Find the length of R S. S cm in. 1 2 The long marks are centimeters, and the shorter marks are millimeters. The length of M N is 3.4 centimeters. The measurement is accurate to within 0.5 millimeter, so M N is between 3.35 centimeters and 3.45 centimeters long. Exercises The long marks are inches and the short marks are quarter inches. The length of R S is about inches. The measurement is accurate to within one half of a quarter inch, or 1 8 inch, so R S is between 1 5 inches and Find the length of each line segment or object. inches long. Lesson S T cm in in. 1 2 cm Find the precision for each measurement in mm cm 8. 2 ft mm yd 2 Glencoe/McGraw-Hill 7 Glencoe Geometry

10 1-2 NME TE ERIO Study Guide and Intervention (continued) Linear Measure and recision alculate Measures On Q, to say that point M is between points and Q means, Q, and M are collinear and M MQ Q. On, 3 cm. We can say that the segments are congruent, or. Slashes on the figure indicate which segments are congruent. M Q Example 1 Example 2 Find EF. Find x and. E 1.2 cm 1.9 cm alculate EF by adding E and F. E F EF EF 3.1 EF Therefore, E F is 3.1 centimeters long. F 2x 5 x 2x is between and. x 2x 2x 5 3x 2x 5 x 5 2x 5 2(5) 5 15 Exercises Find the measurement of each segment. ssume that the art is not drawn to scale. 1. R T cm 2.5 cm R S T in. 6 in. 3. X Z 3 4. W X 31 2 in. 4 in. X Y Z W 6 cm X Y Find x and RS if S is between R and T. 5. RS 5x, ST 3x, and RT RS 2x, ST 5x 4, and RT RS 6x, ST 12, and RT RS 4x, R S S T, and RT 24. Use the figures to determine whether each pair of segments is congruent. 9. and 11 cm 10. X Y and Y Z X 5 cm 5 cm 11 cm 3x 5 5x 1 Y 9x 2 Z Glencoe/McGraw-Hill 8 Glencoe Geometry

11 1-2 NME TE ERIO Skills ractice Linear Measure and recision Find the length of each line segment or object cm in. 1 2 Find the precision for each measurement feet centimeters inches 2 Find the measurement of each segment. 6. N Q G H Q 1in in. N 4.9 cm 5.2 cm F 9.7 mm G H 15 mm Lesson 1-2 Find the value of the variable and YZ if Y is between X and Z. 9. XY 5p, YZ p, and XY XY 12, YZ 2g, and XZ XY 4m, YZ 3m, and XZ XY 2c 1, YZ 6c, and XZ 81 Use the figures to determine whether each pair of segments is congruent. 13. E, 14. M, N 15. W X, W Z 2 m 9 ft 12 yd Y Z 3 m 3 m M 10 yd 5 ft 5 ft E 5 m 12 yd N X W Glencoe/McGraw-Hill 9 Glencoe Geometry

12 1-2 NME TE ERIO ractice Linear Measure and recision Find the length of each line segment or object. 1. E F 2. in. 1 2 cm Find the precision for each measurement meters inches millimeters 4 Find the measurement of each segment. 6. S W X 18.4 cm Q 4.7 cm S in in. W X 89.6 cm Y 100 cm Find the value of the variable and KL if K is between J and L. 9. JK 6r, KL 3r, and JL JK 2s, KL s 2, and JL 5s 10 Use the figures to determine whether each pair of segments is congruent. 11. T U, S W 12., 13. G F, F E T 2 ft S 12.7 in. 5x G H 2 ft 3 ft 6x U W 3 ft 12.9 in. F E 14. RENTRY Jorge used the figure at the right to make a pattern for a mosaic he plans to inlay on a tabletop. Name all of the congruent segments in the figure. F E Glencoe/McGraw-Hill 10 Glencoe Geometry

13 NME TE ERIO 1-2 Reading to Learn Mathematics Linear Measure and recision re-ctivity Why are units of measure important? Read the introduction to Lesson 1-2 at the top of page 13 in your textbook. The basic unit of length in the metric system is the meter. How many meters are there in one kilometer? o you think it would be easier to learn the relationships between the different units of length in the customary system (used in the United States) or in the metric system? Explain your answer. Reading the Lesson 1. Explain the difference between a line and a line segment and why one of these can be measured, while the other cannot. 2. What is the smallest length marked on a 12-inch ruler? What is the smallest length marked on a centimeter ruler? 3. Find the precision of each measurement. a. 15 cm b cm Lesson Refer to the figure at the right. Which one of the following statements is true? Explain your answer. 4.5 cm 4.5 cm 5. Suppose that S is a point on V W and S is not the same point as V or W.Tell whether each of the following statements is always, sometimes, or never true. a. VS SW b. S is between V and W. c. VS VW SW Helping You Remember 6. good way to remember terms used in mathematics is to relate them to everyday words you know. Give three words that are used outside of mathematics that can help you remember that there are 100 centimeters in a meter. Glencoe/McGraw-Hill 11 Glencoe Geometry

14 1-2 NME TE ERIO Enrichment oints Equidistant from Segments The distance from a point to a segment is zero if the point is on the segment. Otherwise, it is the length of the shortest segment from the point to the segment. figure is a locus if it is the set of all points that satisfy a set of conditions. The locus of all points that are 1 4 inch from the segment is shown by two dashed segments with semicircles at both ends. 1. Suppose,,, and are four different points, and consider the locus of all points x units from and x units from. Use any unit you find convenient. The locus can take different forms. Sketch at least three possibilities. List some of the things that seem to affect the form of the locus. X Y X Y R S Q 2. onduct your own investigation of the locus of points equidistant from two segments. escribe your results on a separate sheet of paper. Glencoe/McGraw-Hill 12 Glencoe Geometry

15 1-3 NME TE ERIO Study Guide and Intervention istance and Midpoints istance etween Two oints istance on a Number Line istance in the oordinate lane a b b a or a b ythagorean Theorem: a 2 b 2 c 2 istance Formula: d (x x 2 1 ) 2 (y 2 y 1 ) 2 ( 2, 1) O y (1, 3) x (1, 1) Example 1 Example 2 Find ( 4) Find the distance between ( 2, 1) and (1, 3). ythagorean Theorem () 2 () 2 () 2 () 2 (3) 2 (4) 2 () istance Formula d (x x 2 1 ) 2 (y 2 y 1 ) 2 (1 2)) ( 2 (3 ( 1)) 2 (3) 2 (4) Exercises Use the number line to find each measure G 3. F 4. EF 5. G 6. G 7. E 8. E E F G Lesson 1-3 Use the ythagorean Theorem to find the distance between each pair of points. 9. (0, 0), (6, 8) 10. R( 2, 3), S(3, 15) 11. M(1, 2), N(9, 13) 12. E( 12, 2), F( 9, 6) Use the istance Formula to find the distance between each pair of points. 13. (0, 0), (15, 20) 14. O( 12, 0), ( 8, 3) 15. (11, 12), (6, 2) 16. E( 2, 10), F( 4, 3) Glencoe/McGraw-Hill 13 Glencoe Geometry

16 1-3 NME TE ERIO Study Guide and Intervention (continued) istance and Midpoints Midpoint of a Segment Midpoint on a If the coordinates of the endpoints of a segment are a and b, Number Line then the coordinate of the midpoint of the segment is a. b 2 If a segment has endpoints with coordinates (x Midpoint on a 1, y 1 ) and (x 2, y 2 ), oordinate lane then the coordinates of the midpoint of the segment are x 1 x, 2 y 1 y Example 1 Q Find the coordinate of the midpoint of Q The coordinates of and Q are 3 and 1. If M is the midpoint of Q, then the coordinate of M is or Example 2 M is the midpoint of Q for ( 2, 4) and Q(4, 1). Find the coordinates of M. x 2 M x 1, y y , or (1, 2.5) Exercises Use the number line to find the coordinate of the midpoint of each segment. 1. E 2. G E F G F 4. E G G E Find the coordinates of the midpoint of a segment having the given endpoints. 9. (0, 0), (12, 8) 10. R( 12, 8), S(6, 12) 11. M(11, 2), N( 9, 13) 12. E( 2, 6), F( 9, 3) 13. S(10, 22), T(9, 10) 14. M( 11, 2), N( 19, 6) Glencoe/McGraw-Hill 14 Glencoe Geometry

17 1-3 NME TE ERIO Skills ractice istance and Midpoints Use the number line to find each measure. J K L M N 1. LN 2. JL KN 4. MN Use the ythagorean Theorem to find the distance between each pair of points y y S G O x O x F 7. K(2, 3), F(4, 4) 8. ( 3, 1), Q( 2, 3) Use the istance Formula to find the distance between each pair of points. 9. Y(2, 0), (2, 6) 10. W( 2, 2), R(5, 2) 11. ( 7, 3), (5, 2) 12. ( 3, 1), Q(2, 6) Use the number line to find the coordinate of the midpoint of each segment. 13. E 14. E Lesson Find the coordinates of the midpoint of a segment having the given endpoints. 17. T(3, 1), U(5, 3) 18. J( 4, 2), F(5, 2) Find the coordinates of the missing endpoint given that is the midpoint of N Q. 19. N(2, 0), (5, 2) 20. N(5, 4), (6, 3) 21. Q(3, 9), ( 1, 5) Glencoe/McGraw-Hill 15 Glencoe Geometry

18 1-3 NME TE ERIO ractice istance and Midpoints Use the number line to find each measure. 1. VW 2. TV 3. ST 4. SV S T U V W Use the ythagorean Theorem to find the distance between each pair of points. 5. y 6. Z y S M O x O x E Use the istance Formula to find the distance between each pair of points. 7. L( 7, 0), Y(5, 9) 8. U(1, 3), (4, 6) Use the number line to find the coordinate of the midpoint of each segment. 9. R T 10. Q R Q R S T S T 12. R Find the coordinates of the midpoint of a segment having the given endpoints. 13. K( 9, 3), H(5, 7) 14. W( 12, 7), T( 8, 4) Find the coordinates of the missing endpoint given that E is the midpoint of F. 15. F(5, 8), E(4, 3) 16. F(2, 9), E( 1, 6) 17. ( 3, 8), E(1, 2) 18. ERIMETER The coordinates of the vertices of a quadrilateral are R( 1, 3), S(3, 3), T(5, 1), and U( 2, 1). Find the perimeter of the quadrilateral. Round to the nearest tenth. Glencoe/McGraw-Hill 16 Glencoe Geometry

19 1-3 NME TE ERIO Reading to Learn Mathematics istance and Midpoints re-ctivity How can you find the distance between two points without a ruler? Reading the Lesson Read the introduction to Lesson 1-3 at the top of page 21 in your textbook. Look at the triangle in the introduction to this lesson. What is the special name for in this triangle? Find in this figure. Write your answer both as a radical and as a decimal number rounded to the nearest tenth. 1. Match each formula or expression in the first column with one of the names in the second column. a. d (x x 2 1 ) 2 ( y 2 y 1 ) 2 b. a b 2 i. ythagorean Theorem ii. istance Formula in the oordinate lane c. XY a b iii. Midpoint of a Segment in the oordinate lane d. c 2 a 2 b 2 iv. istance Formula on a Number Line x 2 y 2 e. x 1, y v. Midpoint of a Segment on a Number Line 2. Fill in the steps to calculate the distance between the points M(4, 3) and N( 2, 7). Let (x 1, y 1 ) (4, 3). Then (x 2, y 2 ) (, ). d ( ) 2 ( ) 2 MN ( ) 2 ( ) 2 MN ( ) 2 ( ) 2 MN MN Find a decimal approximation for MN to the nearest hundredth. Lesson 1-3 Helping You Remember 3. good way to remember a new formula in mathematics is to relate it to one you already know. If you forget the istance Formula, how can you use the ythagorean Theorem to find the distance d between two points on a coordinate plane? Glencoe/McGraw-Hill 17 Glencoe Geometry

20 1-3 NME TE ERIO Enrichment Lengths on a Grid Evenly-spaced horizontal and vertical lines form a grid. You can easily find segment lengths on a grid if the endpoints are grid-line intersections. For horizontal or vertical segments, simply count squares. For diagonal segments, use the ythagorean Theorem (proven in hapter 7). This theorem states that in any right triangle, if the length of the longest side (the side opposite the right angle) is c and the two shorter sides have lengths a and b, then c 2 a 2 b 2. Find the measure of E F on the grid at the right. Locate a right triangle with E F as its longest side. E 2 Example 5 F EF units I J E R S Q L F K N M Find each measure to the nearest tenth of a unit. 1. I J 2. M N 3. R S 4. Q S 5. I K 6. J K 7. L M 8. L N Use the grid above. Find the perimeter of each triangle to the nearest tenth of a unit QRS 11. EF 12. LMN 13. Of all the segments shown on the 14. On the grid, 1 unit 0.5 cm. How can the grid, which is longest? What is its answers above be used to find the measures length? in centimeters? 15. Use your answer from exercise 8 to 16. Use a centimeter ruler to find the perimeter calculate the length of segment LN of triangle IJK to the nearest tenth of a in centimeters. heck by measuring centimeter. with a centimeter ruler. Glencoe/McGraw-Hill 18 Glencoe Geometry

Geometry. Chapter 1 Resource Masters

Geometry. Chapter 1 Resource Masters Geometry hapter esource Masters eading to Learn Mathematics Vocabulary uilder This is an alphabetical list of the key vocabulary terms you will learn in hapter. s you study the chapter, complete each term