Geometry: A Complete Course

Transcription

1 Geometry: omplete ourse (with Trigonometry) Module Progress Tests Written by: Larry E. ollins

2 Geometry: omplete ourse (with Trigonometry) Module - Progress Tests opyright 2014 by VideotextInteractive Send all inquiries to: VideotextInteractive P.O. ox Indianapolis, IN ll rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of the publisher, Printed in the United States of merica. ISN RPInc

3 Table of ontents Progress Tests Unit II - Fundamental Terms Part - Undefined Terms LESSON 1 - In lgebra combined LESSON 2 - In Geometry Quiz Quiz Part - Defined Terms LESSON 1 - Good Definitions Quiz Quiz LESSON 2 - Definitions about Points Quiz Quiz LESSON 3 - Definitions about Lines LESSON 4 - Definitions about Rays combined LESSON 5 - Definitions about Line Segments Quiz Quiz LESSON 6 - Definitions about ngles as Sets of Points LESSON 7 - Definitions about Measurements of ngles combined Quiz Quiz LESSON 8 - Definitions about Pairs of ngles Quiz Quiz LESSON 9 - Definitions about ircles Quiz Quiz Module - Table of ontents i

4 Part - Postulates (or xioms) LESSON 1 - Need Quiz Quiz LESSON 2 - Postulate 1 - Existence of Points LESSON 3 - Postulate 2 - Uniqueness of Lines, Planes, and Spaces combined LESSON 4 - Postulate 3 - One, Two, and Three Dimensions Quiz Quiz LESSON 4 - Postulate 3 - One, Two, and Three Dimensions LESSON 5 - Postulate 4 - Separation of Lines, Planes, and Spaces combined LESSON 6 - Postulate 5 - Intersection of Lines or Planes Quiz Quiz LESSON 7 - Postulate 6 - Ruler combined LESSON 8 - Postulate 7 - Protractor Quiz Quiz LESSON 9 - Postulate 8 - ircle LESSON 10 - Postulate 9 - Parallel Lines LESSON 11 - Postulate 10 - Perpendicular Lines combined Quiz Quiz Unit II Test - Form Unit II Test - Form Unit II Test - Form Unit II Test - Form D ii Module - Table of ontents

5 Quiz Form lass Date Score Unit II - Fundamental Terms Part - Undefined Terms Lesson 1 - In lgebra Lesson 2 - In Geometry From the list of five mathematical parts of speech and four types of mathematical expressions, identify each of the items in problems 1 through 9 below, using the most appropriate term. 1. 7n The fraction bar in x 3 = 7 6. The fraction bar in = The m in 2m (8)(3) VideoTextInteractive Geometry: omplete ourse 5

6 Unit II, Part, Lesson 2, Quiz Form ontinued 3. Refer to the figure at the right to identify each of the following sets of points as coplanar or non-coplanar. E D F a) F, G,, and b) D,,, and c), G,, and d),, E, and D G 4. For each of the given sets of numbers, which number is between the other two? a) { 18., ( 13. ) 2, 3 } b) ,, VideoTextInteractive Geometry: omplete ourse

7 Unit II, Part, Lesson 2, Quiz Form ontinued 3. Refer to the figure at the right to identify each of the following sets of points as coplanar or non-coplanar. E D G F J a) F, G, E, and b) D, F, G, and c), J, E, and D d), F, D, and E 4. For each of the given sets of numbers, which number is between the other two? a) 2 3,, b) {. 615,. 636,. 513 } VideoTextInteractive Geometry: omplete ourse

8 Unit II, Part, Lesson 3,4,5, Quiz Form ontinued 9. Is it true that the union or intersection of two sets of points on a line must be a subset of the line? (yes or no) Explain your answer. 10. For each of the following, if you are given the coordinates c, d, s, and t, for the points, D, S, and T respectively, determine if D ST. Show your work. c d s t a) b) c) a b a + b 3b 5b 11. Find the coordinate of the midpoint of a line segment, if the coordinates of the end points are 5x 2 and 13x 2. (Hint: Think about a number line) 12. What is meant if we write = D? VideoTextInteractive Geometry: omplete ourse

9 Unit II, Part, Lesson 3,4,5, Quiz Form ontinued 9. What is the basic difference between a ray and each of the following? a) line segment b) half line c) line 10. For each of the following, if you are given the coordinates c, d, s, and t, for the points, D, S, and T respectively, determine if D ST? Show your work. c d s t a) b) c) VideoTextInteractive Geometry: omplete ourse

10 Unit II, Part, Lesson 3,4,5, Quiz Form ontinued 11. = 7x 2, = 2x + 8, and is the midpoint of. Find the values of x,,, and. 12. May we write = D? Explain your answer VideoTextInteractive Geometry: omplete ourse 21

11 Quiz Form lass Date Score Unit II - Fundamental Terms Part - Defined Terms Lesson 6 - Definitions about ngles as Sets of Points Lesson 7 - Definitions about Measurement of ngles For exercises 1 through 5, refer to the angle at the right. 1. the angle in two ways. 2. the two sides of the angle. 3. the vertex of the angle. G D F 4. Point F is in the of the angle. 5. Point G is in the of the angle. For exercises 6 through 9, refer to the figure at the right. 6. How many angles are there, in the figure? U V 7. the angles in the figure. 8. the vertex of the angle. W X 9. Is WV totally in the interior of angle UWX? Explain your answer. 10. In defining an angle, is it possible that the rays forming the angle might have different endpoints? 11. May a page of a book serve as a model of a half plane? perfect model of a half-plane? Explain why or why not VideoTextInteractive Geometry: omplete ourse 23

12 Quiz Form lass Date Score Unit II - Fundamental Terms Part - Defined Terms Lesson 8 - Definitions about Pairs of ngles omplete each statement in exercises 1 through 6, using the figure at the right. 1. HIJ and are vertical angles. 2. MQL and are adjacent angles. H K L J I 3. JIM and are supplementary angles. 4. m MUJ =. T U Q M 5. The complement of TUP is. P 6. KLT forms a linear pair with. In exercises 7 and 8, explain why the angles in each pair are not adjacent. 7. RXT and TQP 8. and D X Q P T R D 9. What is the measure of the supplement, of the complement, of a 35 degree angle? (Hint: Make a drawing) 2014 VideoTextInteractive Geometry: omplete ourse 27

13 Unit II, Part, Lesson 8, Quiz Form ontinued D 60 o 60 o E F Use the figures above to determine whether each statement in problems 10 through 15 is true or false. 10. DEF = 60 O 11. m = 60 O 12. = DEF 13. m m DEF 14. m DEF = 60 O 15. DEF 16. MN bisects PMQ m PMN = 3x + 8 m QMN = 5x 10 M P N Q Find the value of x, and the measure of each angle VideoTextInteractive Geometry: omplete ourse

14 Unit II, Part, Lesson 8, Quiz Form ontinued F 120 o 120 o D E The statements in problems 10 through 15 refer to the figures above. The statements are false. Rewrite each statement in the correct form, in the blank to the right. 10. = 120 O 11. = DEF 12. m m DEF 13. DEF 120 O 14. m 120 O 15. DEF = 16. XQ bisects MXN 1 m MXQ = x m NXQ = 70 M X Q N Find the value of x, and the measure of MXQ VideoTextInteractive Geometry: omplete ourse

15 Quiz Form lass Date Score Unit II - Fundamental Terms Part - Defined Terms Lesson 9 - Definitions about ircles In exercises 1 through 6, refer to the diagram at the right to determine whether the arc is a minor arc, a major arc or a semi circle of circle O. 1. RQ ) 2. MRP ) N P 3. MNQ ) 4. NP ) M O Q 5. RN ) 6. RQP ) R 7. In the figure for exercises 1 through 6, NOP is called a angle. 8. In the figure for exercises 1 through 6, OQ is a of the circle, RP is a of the circle, point O is the of the circle, and RQ is called a of the circle. 9. Tell what fractional part of a circle is represented by the following arcs: a) an arc of 36 O b) an arc of 160 O If an arc is of a circle, it measures degrees VideoTextInteractive Geometry: omplete ourse 31

16 Unit II, Part, Lesson 9, Quiz Form ontinued In exercise 11 through 18, refer to the figure at the right to determine the measure of each arc or angle. 11. m UPX 12. mxw ) U V 13. m VPU 14. mvx ) 15. mvw ) 16. m UVW ) 17. muv ) 18. m VWX ) X P 60 o 150 o W VideoTextInteractive Geometry: omplete ourse

17 Quiz Form lass Date Score Unit II - Fundamental Terms Part - Defined Terms Lesson 9 - Definitions about ircles 1. If two arcs of a circle are equal, are their central angles equal? 2. If an arc of a circle is halved, is its central angle halved? In exercises 3 through 10, refer to the diagram at the right. 3. four central angles of circle P. H 100 o P F G 4. three minor arcs of circle P. E 60 o 60 o 5. two major arcs of circle P. 6. a semi circle of circle P. 7. a pair of arcs you think are congruent. 8. What is m? ) 9. What is mg? ) 10.What is me? ) 2014 VideoTextInteractive Geometry: omplete ourse 33

18 Unit II, Part, Lesson 9, Quiz Form ontinued In exercises 11 through 14, refer to the diagram at the right to find the measures of the indicated arcs. 11. md = ) 2x+35 2x 3x x+45 D 12. m = ) 13. md = ) 14. md = ) VideoTextInteractive Geometry: omplete ourse

19 Unit II, Part, Lesson 9, Quiz Form ontinued In exercises 15 through 18, refer to circle P to find the value of x. Then find the measure of the labeled arcs x o P P x o x o D (5x) o (2x+30) o U (3x) o V M (5x) o Y P (8x) o X P x 60o o Y (8x) o W N 2014 VideoTextInteractive Geometry: omplete ourse 35

20 Unit II, Part, Lesson 1, Quiz Form ontinued 4. State the missing reason (mathematical property) for each step in the following proof. Step 1 Step 2 Step x = = 4 1 x = x x = 4 9 ssumed to be true Step 4 x = Which of the following statements are true for all real numbers a and b? If false, give a numerical example to show it is false. a) If a < b, then a + b < 0. b) If a < b, then a b > VideoTextInteractive Geometry: omplete ourse

21 Quiz Form lass Date Score Unit II - Fundamental Terms Part - Postulates (or xioms) Lesson 1 - Need 1. the property of equality (reflexive, symmetric, or transitive) that allows you to conclude that each statement is true. a) MN = MN b) If x = 3, then 3 = x c) If m R = m S and m S = m T, then m R = m T 2. Write a paragraph proof to show that the following is true: x If - 9 = 16, then x = Use the given property in each of the following to draw a conclusion. a) Transitive Property: If a = b and b = c, then a = c m = m, m = 61 onclusion: b) Symmetric Property: If m = n, then n = m. = D onclusion: c) Multiplication Property: If a = b and c = d, then a c = b d 4 ( x ) = 52, 4 =.25 onclusion: 2014 VideoTextInteractive Geometry: omplete ourse 39

22 Unit II, Part, Lesson 1, Quiz Form ontinued 4. State the missing reason (mathematical property) for each step in the following proof. Step 1 Step 2 Step 3 2 x <8 1 2 ( ) x > x > 4 2 > x ssumed to be true Step 4 x > 4 5. Which of the following statements are true for all real numbers a and b? If false, give a numerical example to show it is false. a) If a < b, then a b > 0. b) If a < b, then a b > 0. (b 0) VideoTextInteractive Geometry: omplete ourse

23 Quiz Form lass Date Score Unit II - Fundamental Terms Part - Postulates (or xioms) Lesson 2 - Postulate 1 (Existence of Points) Lesson 3 - Postulate 2 (Uniqueness of Lines, Planes and Spaces) Lesson 4 - Postulate 3 (One, Two, and Three Dimensions) In the given figure, Plane M contains point D, and collinear points,, and. Point E is not in Plane M. Give a reason for each statement in problems 1-5. E 1. There is exactly one plane containing points E,, and. M D 2. There is exactly one line containing points D and. 3. Line D is in Plane M. 4. Point D is not on. 5. Point D is not on plane E VideoTextInteractive Geometry: omplete ourse 43

24 Unit II, Part, Lessons 2, 3& 4, Quiz Form ontinued In the given figure, plane M intersects plane N. Use this information to answer problems two points that determine line. 7. three points that determine plane M. M 8. the intersection of plane M and plane N. 9. Does D lie on plane M?. D E l N 10. Does plane N contain any points not on?. 11. Only one contains and point D. 12. The plane containing points,, and also contains point. 13. Line E and line intersect at point. 14. Line E and line determine plane. In problems 15-18, tell how many planes at most can pass through each set of points. 15. One line 16. Two points 17. Three collinear points 18. line and a point not on the line VideoTextInteractive Geometry: omplete ourse

25 Unit II, Part, Lessons 4,5&6, Quiz Form ontinued To answer problems 9-12, you will have to visualize certain lines and planes not drawn specifically in the diagram. 9. a plane that contains. E H D G F 10. the intersection of plane DFE and plane D. 11. two lines that are not shown in the diagram and that do not intersect. 12. three planes that do not intersect EF. In the figure at the right, line separates plane M into two half-planes. Use this information to answer problem 13. l 13. Point and point are in different half-planes. Justify this statement by using the appropriate part of the line separation postulate. M VideoTextInteractive Geometry: omplete ourse

26 Quiz Form lass Date Score Unit II - Fundamental Terms Part - Postulates (or xioms) Lesson 7 - Postulate 6 (Ruler) Lesson 8 - Postulate 7 (Protractor) 1. omplete the following relations: a) 7 = c) 0 = e) = b) -7 = d) 6-9 = f) 9 ( -6) = l 2. Using the given number line, find the distance between each indicated pair of points D E F a) and b) and c) F and D d) and F e) and D f) E and g) D and h) and F 3. How many different points can be either to the right or to the left of a given point on a horizontal number line? 2014 VideoTextInteractive Geometry: omplete ourse 53

27 Unit II, Part, Lessons 7& 8, Quiz Form ontinued 4. Suppose on line t, point P corresponds to the real number 2. How many different points exist on t such that the distance from P to each of these points is 12? What real numbers correspond to these points? How many different points exist on t such that the distance from P to each of these points is a given real number q? What are the coordinates of these points? 5. If = 6 and = 38, is necessarily 32? Explain your answer. 6. In the given diagram of collinear points, PT = 20, QS = 6, and PQ = QR = RS. Find each length asked for below. P Q R S T a) QR b) ST c) RT d) SP 7. Measure the given line segment to the nearest millimeter. D D = 8. Use a protractor to determine the measure of each angle named below. N a) m NOP = b) m NOM = P c) m QON = M O Q VideoTextInteractive Geometry: omplete ourse

28 Unit II, Part, Lessons 7& 8, Quiz Form ontinued 9. If M is between L and N and LM = 1, find x, LM, MN and LN by 2 x +2, MN=3 x +3 2, and LN = 5 x + 2 using the Segment ddition Postulate. 10. Explain why the ngle ddition Property begins with If a ray O lies between rays O and O,... Use a sketch of O with ray O not between rays O and O to illustrate what the relationship would be otherwise VideoTextInteractive Geometry: omplete ourse 55

29 Quiz Form lass Date Score Unit II - Fundamental Terms Part - Postulates (or xioms) Lesson 7 - Postulate 6 (Ruler) Lesson 8 - Postulate 7 (Protractor) 1. omplete the following: 3 a) 0-9 = c) e) 4 = = ( )( ) -13 = b) - 5+3= d) = f) 2. Find the distance between each pair of points having the given coordinates. a) 3 and 10 b) 2.4 and 3.1 c) 7 1 d) 3.14 and and e) f) g) 11.3 and 7.1 h) a and b 8 and 1 7 and Suppose line t contains M and Q. Does a point P always exist such that Q is between M and P? Does a point R always exist such that M is between R and Q? What in general does this tell us? 2014 VideoTextInteractive Geometry: omplete ourse 57

30 Unit II, Part, Lessons 7& 8, Quiz Form ontinued 4. How many different points on a line can be half the distance from one given point to another given point? Explain your answer. 5. If,, and are collinear, = 26, and = 16, is necessarily 42? Explain your answer. 6. In the accomplanying diagram of collinear points, PT = 26, PR = 8, and PQ = QR = RS. Find each length asked for. P Q R S T a) RS b) PQ c) QS d) QT 7. Measure the given line segment to the nearest millimeter. = 8. Use a protractor to determine the measure of each angle named below. a) m GOI = H I b) m HOJ = c) m IOH = G O J VideoTextInteractive Geometry: omplete ourse

31 Unit II, Part, Lessons 7& 8, Quiz Form ontinued 9. If M is between L and N and LM = 1 x, find x, LM, MN and LN 3 +4, MN=4 x +4 3, and LN = 4 x by using the Segment ddition Postulate. 10. What requirement for adding segment lengths is similar to the requirement that a ray is between two rays when adding angles? 2014 VideoTextInteractive Geometry: omplete ourse 59

32 Unit II, Part, Lessons 9, 10 &11, Quiz Form ontinued Using the information on the circle at the right, find the distance between each pair of points in problems 3-7. Give two distance measurements in each case. 180 E D to H 4. G to F H Minor rc Major rc Minor rc Major rc G D to 6. F to D 7. to Minor rc Major rc Minor rc Major rc Minor rc Major rc 8. How many lines can be perpendicular to a given plane from a given line not in the given plane? 9. How many lines can be perpendicular to a given line at a given point on the given line? VideoTextInteractive Geometry: omplete ourse

33 Unit II, Part, Lessons 9, 10 &11, Quiz Form ontinued Using the given circle, find the distance between the following pairs of points. Give two distance measurements in each case. D D to E 4. to H 180 E 0 Minor rc Major rc Minor rc Major rc 253 F G H 5. F to G 6. to E 7. D to G Minor rc Major rc Minor rc Major rc Minor rc Major rc l In the given figure, line passes through points R and S. point is not on line. points R, S, and are in plane M. Use this information to solve problems 8 and 9 below. 8. State the postulate that guarantees the existence of only one line through point, which is perpendicular to line. R S l l M 9. Trace the figure in Exercise 8, and complete the picture by sketching the perpendicular line through point to line VideoTextInteractive Geometry: omplete ourse

34 Unit II, Fundamental Terms, Unit Test Form ontinued, Page 6 In problems 44-47, classify each statement as true or false. (-4) 44. It is possible for two intersecting lines to be non-coplanar. (-4) 45. It is possible to locate three points in such a position that an unlimited number of planes contain all three points. (-4) 46. Through any three points there is at least one line. (-4) 47. If points and lie in plane P, then so does any point of. For problems 48-51, use a protractor to draw angles having the following degree measures O O O O VideoTextInteractive Geometry: omplete ourse

35 Unit II, Fundamental Terms, Unit Test Form ontinued, Page 7 In the figure at the right, the following pairs of angles are complementary: O and O, OD and DOE, EOF and FOG, GOH and HO. If m O = 45 O, m OD = 30 O, EOF FOG, and m HO = 10 O, use this information to find the measures asked for in problems m O 53. m DOE 54. m GOH 55. m GOF 56. m FOE O P Q G F H O E D In the figure at the right, XY and are diameters.use the information given in the figure to solve problems Y (-9) 57. Find the value of x. P (3x) O O (3x-3) O (2x+15) X (-9) 58. Find m PY 59. Find myx (-9) 60. Find m YP 61. Find m P (-9) 62. Find mx 63. Find m P (-9) ( ( 64. Find m XP 65. Find m (-9) 66. Find m (-9) (-9) (-9) (-9) ( ( 2014 VideoTextInteractive Geometry: omplete ourse 71

36 Unit II, Fundamental Terms, Unit Test Form ontinued, Page 8 (-11) 67. Draw triangle in which angle is an obtuse angle. Then, through point, sketch the perpendicular line to line. (-10) 68. Draw triangle in which angle is an obtuse angle. Then, through point, sketch the line parallel line to line. (-5) 69. Plane M is divided into two half planes by line. Point and Point are in different half planes. Explain why we know this is true from the perspective of the separation postulate. M l VideoTextInteractive Geometry: omplete ourse

37 Unit II, Fundamental Terms, Unit Test Form ontinued, Page 5 For exercises 44-48, refer to a numberline, not pictured here, with the following information. Point has coordinate 2 and point has coordinate (-5) 44. What is the length of. (-5) 45. What is the coordinate of the midpoint of? (-5) 46. If is the midpoint of P, what is the coordinate of point P? (-5) 47. What is the coordinate of a point that is on and is 4 units from. (-5) 48. What is the coordinate of a point that is 4 units from point, but is not on? (-2) 49. Is it possible for a line and a point to be non-coplanar? (-5) 50. Is it possible for the intersection of two planes to consist of a segment? In the figure at the right, VR bisects QVS, m PVQ = 72 O, m TVS = 70 O, and VP and VT are opposite rays. Use this information to find the measures asked for in problems m QVS 52. m QVR 53. m PVR P Q R S 54. m TVR 55. m PVS V T 56. m TVQ 2014 VideoTextInteractive Geometry: omplete ourse 79

38 Unit II, Fundamental Terms, Unit Test Form ontinued, Page 6 Use the figure at the right to find the measures asked for in problems m O D 53. m EOF E F O m DOE 55. m OD 56. m EO Using the diagram at the right, and your powers of observation, determine if each arc in problems is a minor arc, major arc, or semicircle. (-9) 57. DE ( M (-9) 58. E ( E D (-9) 59. E ( ( (-9) 60. D VideoTextInteractive Geometry: omplete ourse

39 Unit II, Fundamental Terms, Unit Test Form D ontinued, Page 2 (-8) 13. linear pair (-2) 14. line (-1) 15. Multiplicative Inverse Property (-1) 16. Distributive Property of Multiplication Over ddition (-9) 17. central angle (-6) 18. If two different lines intersect, then the intersection is... (-1) 19. ommutative Property of ddition (-4) 20. For every plane in space, the is at least one point in space... (-8) 21. congruent angles (-9) 22. semi circle (-6) 23. dihedral angle n) set of points, not all of which lie on the same line. o) The union of two half planes with the same edge. p) a + b = b + a 1 q) of complete rotation 4 r) Mathematical expression which contains a relation symbol, but does not contain a placeholder symbol. s) Two angles with the same measure. t) unique line. u) There exists a unique real number 0, such that for every real number a, a 0 = 0 v) One of two sets of points on either side of the separation point on a line. w) that is not on the plane. x) a (b + c) = a b + a c y) If a < b, then a + c < b + c z) Symbols that group numbers together, or, more generally, symbols which indicate the associations of mathematics. (-9) 24. minor arc (-9) 25. center (-4) 26. half-line VideoTextInteractive Geometry: omplete ourse