Geometry: omplete ourse (with Trigonometry Module - Student WorkText Written by: Thomas E. lark Larry E. ollins
Geometry: omplete ourse (with Trigonometry Module Student Worktext opyright 2014 by VideotextInteractive Send all inquiries to: VideotextInteractive P.. ox 19761 Indianapolis, IN 46219 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 1-59676-094-X 1 23456 7 8 9 10 - RPInc - 18 17 16 15 14
Table of ontents Unit II - Fundamental Terms Part - Undefined Terms LESSN 1 In lgebra................................................114 LESSN 2 In Geometry..............................................118 Part - Defined Terms LESSN 1 Good Definitions...........................................121 LESSN 2 Definitions bout Points.....................................124 LESSN 3 Definitions bout Lines.....................................128 LESSN 4 Definitions bout Rays......................................131 LESSN 5 Definitions bout Line Segments..............................134 LESSN 6 Definitions bout ngles as Sets of Points......................141 LESSN 7 Definistions bout Measurement of ngles......................147 LESSN 8 Definitions bout Pairs of ngles.............................151 LESSN 9 Definitions bout ircles....................................157 Part - Postulates (or xioms LESSN 1 Need....................................................165 LESSN 2 Postulate 1: Existence of Points...............................170 LESSN 3 Postulate 2: Uniqueness of Lines, Planes, and Spaces..............175 LESSN 4 Postulate 3: ne, Two, and Three Dimensions...................178 LESSN 5 Postulate 4: Separation of Lines, Planes and Spaces...............180 LESSN 6 Postulate 5: Intersection of Lines or Planes......................184 LESSN 7 Postulate 6: Ruler..........................................187 LESSN 8 Postulate 7: Protractor.......................................194 LESSN 9 Postulate 8: ircle..........................................198 LESSN 10 Postulate 9: Uniqueness of Parallel Lines.......................202 LESSN 11 Postulate 10: Uniqueness of Perpendicular Lines.................207 ppendices ppendix Properties of Real Numbers...................................1 ppendix Definitions and Important Terms...............................1 Module - Table of ontents i
Lesson 1 Exercises: Each item in the column on the left represents a symbol of mathematics, or an illustration used in Geometry. Match the item with the most appropriate description from the column on the right. (Note: Some of the numbered symbols and illustrations are on the next page. 1. a relation symbol indicating that a given set is a subset of a second set, where the given set is the smaller of the two sets. 2. b n operation symbol which indicates that two or more sets are to be combined into one set, containing all of the elements from both original sets. The symbol is read as the "union". 3. M c relation symbol which indicates that two geometric figures have the same size and shape. 4. 3 2 6 d number symbol (specifically a Roman numeral representing the quantity 7. 2 + ( ( 5. e relation symbol which indicates that geometric figures are parallel. 6. f relation symbol indicating that two numbers are approximately equal to each other. 7. g symbol used to represent a line (read line v 8. h Grouping symbols used to enclose an expression. Specifically, a set of parentheses. 9. VII i number symbol representing the ratio of the circumference of a circle to the diameter of the circle. 10. j n illustration representing a reflection. 11. k relation symbol which is used to state that an object is a member of a set, as in a is a member of the set of vowels. 12.! l Using the more general notion of a number symbol as representing the "things" of mathematics, a symbol used to name a triangle, read "triangle ". 13. = m symbol used to indicate that a statement and its converse are both true. Part Undefined Terms 116
14. n n illustration which indicates that two lines are at right angles to each other. {} 15. o relation symbol indicating that a given set is a subset of a second set, where the two sets are equal. 16. P p n operation symbol indicating the translation of a geometric figure. 17. q Set builder notation illustrating the set of all objects that are elements of set. { } 18. x x Α r n operation symbol which indicates that two or more sets are to be combined into one set, containing only members common to all sets. l 35 19. s symbol used to name a point, read point P. 20. t relation symbol used to indicate that two geometric figures are perpendicular. 21. l u symbol used to name a line. It is read, "line ", where and are two points on the line. M 22. v grouping symbol used to enclose a listed set. N 23. π w n illustration used to indicate or name a plane, much like a number symbol names a number. 24. x n illustration showing the rotation of a point through an angle of 35. 25. iff y relation symbol indicating that two sets are equal. 26. z closed phrase. 117 Unit II Fundamental Terms
Example 3: onsider the following definition: bedroom is a room where people sleep. This is not a good definition. Tell why. Solution: The statement does not satisfy the second of the three qualities of a good definition. It does not state distinguishing properties. Example 4: onsider the following definition: triangle is a polygon made with 3 line segments and 3 angles. This is not a good definition. Tell why. Solution: The statement does not satisfy the third of the three qualities of a good definition. It is not concise. It is not necessary to say 3 angles. (Unit I, Part, Lesson 3 Lesson 1 Exercises: Tell whether each of the following statements in Exercises 1 through 10 is acceptable as a definition. If not, explain why, referring to the three qualities of a good definition. 1. restaurant is a place where food is eaten. 2. finite set is a set containing a definite number of members. 3. pen is an instrument used for writing. 4. n integer is a real number. 5. ruler is a straight-edge with markings for measurement. 6. The empty set is the set having no members. 7. bicycle is a vehicle. 8. late student is a person who arrives after the bell rings. 9. The intersection of two sets of points is the set of points common to both sets of points. 10. pencil is a writing instrument which uses lead. 11. For each of the statements in Exercises 1 through 10 which were considered unacceptable as definitions, restate them as acceptable definitions, if possible. 12. From the perspective of mathematical logic, if a statement is "reversible" (the statement and its converse are both true, must it be a good definition? Why or why not? 13. From the perspective of mathematical logic, if a statement is not "reversible" (the statement and its converse are not both true, may it be a good definition? Why or why not? 123 Unit II Fundamental Terms
In Exercises 29 through 37, refer to the diagram at the right, and name all of the points that are not coplanar with the given points. 29. X, Y and Z 30. W, P and Q 31. P,, and X 32. Q, R, and Y 33. P, R, and 34., X, and Y 35., P, and Z 36. R, W, and Z 37. W, P, and R X R Y W P Z Q For each statement below, in Exercises 38 through 40, indicate which are true and which are false. 38. If point lies between points and, then may lie between and. 39. etween two distinct points R and S, there is always another point. 40. For two distinct points M and N, there is another point Q such that N lies between M and Q. For each of the sets of coordinates given below, in Exercises 41 and 42, which number is between the other two? 41. - 5, -5 2, - 2.4 42..713, 7 11, 3 4 127 Unit II Fundamental Terms
Example 4: Each of the following diagrams illustrates the definition of segment bisector. Identify each bisector by describing the diagram. Point is the midpoint of. a b M M c d E N l e f X Solution: a Point is a bisector of. b M is a bisector of. c Plane M is a bisector of. d E is a bisector of. e NX is a bisector of. f Line is a bisector of. Point, M, plane M, E, NX, and line identify or pass through the midpoint of. Therefore, each is a bisector of the segment. 137 Unit II Fundamental Terms
Unit II Fundamental Terms Part Defined Terms Lesson 6 Definitions bout ngles as Sets of Points bjective: To understand and clearly define fundamental terms about angles, in our Geometry. Important Terms: ngle From the Latin angulus, meaning, corner, this is, in fact, a corner of a simple closed plane curve, made up of straight line segments. Formally, in our Geometry, a geometric figure is an angle, if and only if, it is the union of two different rays (called the sides of the angle, with a common endpoint (called the vertex of the angle. We refer to an angle by using a small symbol for an angle, along with specific points on the rays, being sure to place the letter for the vertex between the letters for the other two points. See the illustration below. T U V TUV or VUT Half-Line Formally, in our Geometry, a set of points is a half-line, if and only if, it is one of the two sets of points on either side of a specific point on a line, called the separation point. Half-plane set of points is a half-plane, if and only if, it is one of the two sets of points on either side of a specific line on a plane, called the separation line. Half-space set of points is a half-space, if and only if, it is one of the two sets of points on either sides of a specific plane in a space, called the separation plane. Edge The more common name for a separation line in a plane. Generally used when combining a half-plane with a separation line, in an effort to clearly describe the interior and exterior of an angle. Interior of an ngle point can be considered as clearly being in the interior of an angle, only if one ray of the angle lies in the edge of a half-plane, and the point is between the two rays making up the angle. Exterior of an ngle point can be considered as clearly being in the exterior of an angle, only if one ray of the angle lies in the edge of a half-plane, and the point is not between, or on the two rays making up the angle. 141 Unit II Fundamental Terms
3. Refer to the following diagram to complete the following exercises. D a Give another name for D (not D. b The interiors of D and D are both common to the interior of what angle? c Name the angles with point as their vertex. 4. Into how many subsets does an angle separate a plane? Describe these subsets. 5. Is the vertex of an angle in the interior of the angle? In the exterior of the angle? Explain your answers. 6. Is an angle contained in its interior? In its exterior? Explain your answers. 7. Using the figure at right, if one point of is in the interior of D, what do you know about other points of? What would be true of the remaining points of if one point of were in the exterior of D? D 8. re all of the points in the interior of an angle coplanar? re all of the points in the exterior of an angle coplanar? Explain your answers. 9. Refer to the following diagram and answer the following questions. D a What do you know about points of in relation to? b If D is a point in the interior of, what do you know about D? 145 Unit II Fundamental Terms
Lesson 7 Exercises: 1. Refer to the following diagram to find the measure of each angle below. Then classify the angle as acute, right, obtuse, or straight. D E 80 90 100 70 110 60 100 80 120 110 70 50 120 60 130 40 130 50 140 140 40 30 150 30 150 20 160 20 160 10 170 10 170 0 180 0 180 a b HJ c J d FJ e EJ f g D h J i F j H k E l DJ m J F G H J 2. Describe how you know whether to use the outer number scale or the inner number scale on a protractor, when using it to measure an angle. 3. Use your protractor to measure each angle below. Write your answer using correct notation. (i.e. m DEF = Note: In some cases, you may have to extend the rays. T a b U Z V c d Y S T U X H e f M N g h I R J Q P 149 Unit II Fundamental Terms
4. Use your protractor to draw angles with the given measures. Include points so that the names of the angles are correct. a m DEF = 30 b m = 120 c m XYZ = 150 d m MNQ = 45 e m JKL = 60 f m RST = 135 5. Estimate the measure of the angle formed by the hands of a clock, at each of the following times. a 3 o clock b 1 o clock c 5 o clock d 20 minutes past 8 e 6 o clock f 4 o clock 6. Draw a line,. hoose a point between and. Use a protractor to determine how many lines you can draw through that will form a 30 angle with. 7. Just using your intuitive understanding of angle measures, make freehand sketches of angles that approximate the following measures. Use a protractor to check your accuracy. a 60 b 150 c 30 d 45 e 90 f 120 8. Using a ruler, draw a large triangle. Then use a protractor to find the approximate measure of each angle and compute the sum of the three measures. Repeat this exercise once more, but create a triangle with a different shape. Did you get approximately the same result? What might you assume about the sum of the measures of the angles of a triangle? 9. In a coordinate plane, plot the points and sketch DEF. lassify the angle. Write the coordinates of a point that lies in the interior of the angle, and the coordinates of a point that lies in the exterior of the angle. a D ( 3, 2 b D ( 5, 1 c D ( 7, 1 E ( 5, 1 E ( 3, 2 E ( 3, 4 F ( 4, 4 F ( 4, 4 F ( 0, 1 10. Draw five points,,, D, and E so that E is a straight angle, D is a right angle and is a right angle. Part Defined Terms 150
Unit II Fundamental Terms Part Defined Terms Lesson 8 Definitions bout Pairs of ngles bjective: To understand and clearly define fundamental terms about pairs of angles, in our Geometry. Important Terms: ongruent ngles From the Latin meaning, to agree, two angles are said to be congruent, if and only if, they are exactly the same size and shape. This relationship is symbolized by an is equal to sign with a wavy line on top. See the illustration below. MNP Equal ngles Two angles are said to be equal, if and only if, they have the same measure. See the illustration below. = MNP ongruence / Measurement onnection Intuitively, it seems reasonable that two angles which are congruent, will have equal measures. Formally, in our Geometry, two angles are congruent, if and only if, the two angles have equal measures. isector of an ngle From two Latin words meaning, to cut in two, a bisector of an angle is a geometric figure which cuts the angle into two congruent parts. Formally, in our Geometry, a ray is a bisector of an angle, if and only if, it is between the sides of the angle, and divides the angle into two congruent angles. See the illustration below. (Note: Technically a line could also be a bisector. F G H GH bisects FGJ J djacent ngles From a Latin word meaning, to lie near, adjacent angles are angles which lie near each other. Formally, in our Geometry, two angles are adjacent, if and only if, the two angles have a common vertex, and a common ray, and the ray lies between the other sides of the two angles. See the illustration below. (Note: The other two sides, the non-common sides, are called the two exterior sides of the two adjacent angles. Y X Z XY and YZ are adjacent angles 151 Unit II Fundamental Terms
Major rc of a ircle n arc whose points are in the exterior of a central angle of a circle. Formally, in our Geometry, an arc of a circle is a major arc, if and only if, it is the union of two points on the circle (not the endpoints of a diameter, and the set of points on the circle, which lie in the exterior of the angle formed by the radii containing the two points. Note: Remember that a third point on the arc must be used to name a major arc. See the illustration below. D D is a major arc of circle Semicircle From a Latin word meaning half, this is an arc which is of a circle 1 ( 2 of a complete rotation. Formally, in our Geometry, an arc of a circle is a semicircle, if and only if, it is an arc whose endpoints are the endpoints of a diameter of the circle. Note: Since the symbolism for a semicircle will designate two congruent semicircles, it is necessary to use a third point to designate a specific semicircle. See the illustration below. Q 1 2 P R PQR is a semicircle of circle Measure of a Minor rc of a ircle Formally, in our Geometry, the measure of a minor arc of a circle, is equal to the related central angle. See the illustration below. m = m Measure of a Major rc of a ircle Formally, in our Geometry, the measure of a major arc of a circle, is equal to 360, minus, the measure of its related minor arc. See the illustration below. D md = 360 m Measure of a omplete ircle (a complete rotation Related to a central angle of 360 (a complete rotation about the center of a circle, the measure of the arc making up the complete circle, is defined to be 360. 159 Unit II Fundamental Terms
Example 3: Using the diagram at right, classify each line segment named below as a radius, a diameter, or neither. Note: Remember that the term radius and the term diameter can refer to a segment or to the length of a segment. a G d b e E c F f GH E D H Solution: a radius d radius G 8 b radius e diameter c diameter f neither F Example 4: Using the diagram is Example 3, fill in the blanks for each of the following. Solution: a The diameter of circle can be expressed as and. (segment (measure b The radius of circle can be expressed as and. (segment (measure a F (or E and F (or E (or 16 inches b G (or, F,, D,, or E and G (or, F,, D,, or E (or 8 inches Example 5: Using the figures below, find the measure of each arc or angle asked for. Y X 34 W T 151 Z U P 85 V Q R S 48 a m TQR b myzx c m XY d muv e msrt f semicircle UW g mwv h m TQS Solution: a 360 (48 +151 b 360 34 = 326 c 34 (central 360 199 = 161 angle equal to arc d 85 (arc equal to e 360 151 = 209 f 180 central angle g 180 85 = 95 h 151 (central angle equal to arc 161 Unit II Fundamental Terms
Lesson 9 Exercises: 1. Using the figure at the right, determine whether each arc named below is a minor arc, a major arc,or a semicircle of circle. a SU e PQT b QUT f PUQ c QT g PQ d TUP h TUQ P Q U T S 2. Using the figure at the right, find each measure asked for below. J a mkl g m MRL N b mlkn h mkm c mjm i mjmk d mmkn j mjn e mjk k m JRN K R 45 L 55 M f mjml l mmn 3. Use circle Q shown at the right, to find each of the following. a Name two central angles. b Find m. 135 Q c Name a semicircle. d Name two minor arcs. e Name two major arcs. f Find m. Part Defined Terms 162
In Exercises 4 through 14, find the measure of each indicated arc. 126 4. 5. m = 117 mmn = N M Q 6. 7. 125 mptq = mgh = P 91 T G 42 H 8. 9. G X mxy = mfgh = Y H 38 F 10. 11. K J X mx = 115 mkji = M I In Exercises 12 through 14, tell what fractional part of a circle is represented by each given arc measure. 12. 40 13. 30 14. 145 163 Unit II Fundamental Terms
Unit II Fundamental Terms Part Postulates (or xioms Lesson 1 Need bjective: To understand the need for postulates, or axioms, in Mathematics, and to recognize their use, in both lgebra and Geometry. Important Terms: Postulates From a Latin word meaning, demand, these are fundamental statements, or assumptions, which we accept without proof, in Mathematics. They are also called axioms, from the Greek, meaning, that which is thought to be fitting, or worthy. Properties of the Real Numbers This term refers to the postulates, or axioms, which we accepted without proof, in the study of rithmetic. Following is a comprehensive list for your reference. 1. Properties of Relations: Trichotomy - For any real numbers a and b, only one of the following can be true: a = b, a > b, a < b Reflexivity for Equality - For any real number a, a = a Symmetry for Equality - For any real numbers a and b, if a = b, then b = a Transitivity for Equality - For any real numbers a, b, and c, if a = b, and b = c, then a = c Substitution - For any real numbers a, and b, if a = b, then a can be substituted for b in any expression (or b can be substituted for a. Transitivity for Inequality- For any real numbers a, b, and c, if a > b, and b > c, then a > c. Likewise, if a < b, and b < c, then a < c. 2. Properties of Well-Defined operations: Existence - For any real numbers a and b, a + b, a - b, and a b exist. Further, a b exists, as long as b 0 Uniqueness - For any real numbers a and b, a + b, a - b, and a b are unique. Further, a b is unique, as long as b 0 losure - For any real numbers a and b, a + b, a - b, and a b are real numbers. Further, a b is a real number, as long as b 0 3. Properties of perations in General: ommutativity of ddition - For any real numbers a and b, a + b = b + a ommutativity of Multiplication - For any real numbers a and b, a b = b a ssociativity of ddition - For any real numbers a, b, and c, (a + b + c = a + (b + c 165 Unit II Fundamental Terms
ssociativity of Multiplication - For any real numbers a, b, and c, (a b c = a (b c Distributivity of Multiplication over ddition - For any real numbers a, b, and c, a (b + c = a b + a c Distributivity of Multiplication over Subtraction - For any real numbers a, b, and c, a (b - c = a b - a c 4. Properties of perations with Special Numbers: Identity of ddition - There exists a unique real number 0, such that, for every real number a, a + 0 = a Identity of Subtraction - There exists a real number 0, such that, for every real number a, a - 0 = a dditive Inverse - For every real number a, there exists a unique real number, (- a, such that a + (- a = 0 Multiplication by Zero - There exists a unique real number 0, such that, for every real number a, a 0 = 0 Division by Zero - Division by zero is meaningless and undefined. Therefore, it is not allowed. Identity of Multiplication by 1 - There exists a unique real number 1, such that, for every real number a, a 1 = a Identity of Division by 1 - There exists a unique real number 1, such that, for every real number a, a 1 = a Multiplicative Inverse - For every real number a, there exists a unique real 1 1 number,, such that, a = 1 a a Identity of a Power of 1 - There exists a unique real number 1, such that, for every real number a, a 1 = a 5. Properties of perations on Relations: ddition - For any real numbers a, b, and c, if a = b, then a + c = b + c. Likewise, if a > b, then a + c > b + c and, if a < b, then a + c < b + c. Subtraction - For any real numbers a, b, and c, if a = b, then a - c = b - c. Likewise, if a >b, then a - c > b - c and, if a < b, then a - c < b - c. Positive Multiplication - For any real numbers a, b, and c, with c > 0, if a = b, then a c = b c. Likewise, if a > b, then a c > b c and, if a < b, then a c < b c. Positive Division - For any real numbers a, b, and c, with c > 0, a b a b if a = b, then =. Likewise, if a > b, then >, and, if a < b, then a c b c c c < c c Negative Multiplication- For any real numbers a, b, and c, with c < 0, if a > b, then a c < b c, or, if a < b, then a c > b c Negative Division - For any real numbers a, b, and c, a b a with c < 0, if a > b, then <, or, if a < b, then > c c c b c Part Postulates (or xioms 166
5. This statement of the distributive property, a (b + c = a b + a c, is sometimes read, multiplication is distributive over addition. Decide if the following are true. Give numerical examples to defend your answer. a Multiplication is Distributive over Subtraction: a (b c = a b a c b ddition is Distributive over Multiplication: a +(b c = (a + b (a + c c Division is Distributive over ddition: a (b + c = a b + a c d ddition is Distributive over Division: a +(b c = (a + b (a + c 6. State the missing reasons for each step in the following proof: If 4 7x = 25, then x = 3. a 4 7x = 25 a ssumed to be true. b 4 + 4 7x = 4 + 25 b. c 0 7x = 21 c. d 0 + 7x = 21 d. e 7x = 21 e. 1 1 f ( ( 7x = ( 21 f. 7 1 7 7 g ( 7 x = 3 g. h (1 x = 3 h. i x = 3 i. 7. Which of the following statements are true for all real numbers a and b? If a statement is 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. c If a < b, then a b < 0. d If a < b < c, then a + (b c = (a + b (a + c. 169 Unit II Fundamental Terms
Unit II Fundamental Terms Part Postulates (or xioms Lesson 2 Postulate 1: Existence of Points bjective: To understand the assumptions regarding the existence of points, in our Geometry. Important Terms: Postulate 1 - Existence of Points 1st ssumption: Every line contains at least two different points. (Stated as a conditional: If there is a line, then there exist at least two different points on that line. 2nd ssumption: Every plane contains at least three different, non-collinear points. (Stated as a conditional: If there is a plane, then there exist at least three different, non-collinear points on that plane. 3rd ssumption: Space contains at least four different, non-coplanar points, no three of which are collinear. (Stated as a conditional: If there is space, then there exist at least four different, non-coplanar points, no three of which are collinear. Note: Remember. Every geometric figure we will deal with, in our Geometry, is made up of points. Part Postulates (or xioms 170
Example 2: onsider three planes and the relationships that may exist between the planes. How many points must exist in each of the relationships? Sketch a diagram to illustrate each situation. a The planes are parallel. b Two of the planes are parallel and the third plane intersects the two parallel planes. c Each plane intersects the other two planes. d ll three planes intersect in the same line. e ll three planes intersect in a single point. f ll three planes coincide. Solution: a t least 9 points must exist. (Every plane contains at least three different, non-collinear points. M 3 1 2 N 6 4 5 Q 9 7 8 b t least 6 points must exist. This relationship is not unlike considering only two intersecting planes since, if only two of the three planes intersect, the third plane would have to be parallel to one of the two intersecting planes. (Every plane contains at least three different, noncollinear points. Q 5 4 6 3 1 2 M N Plane M PPlane N c t least 5 points must exist. (Every plane contains at least three different, non-collinear points. 1 4 2 3 5 Part Postulates (or xioms 172
Example 2: (continued Solution: (continued d t least 5 points must exist. (Every plane contains at least three different non-coplanar points. 5 2 4 1 3 e Plane M, plane N, and plane Q intersect in point. t least 4 points must exist. M 3 1 4 N 2 Q f t least 3 points must exist. (The reasoning is similar to part e in example 1 about three lines coinciding Example 3: onsider three spaces and the relationships that may exist between the spaces. How many points must exist in each of the relationships between spaces? Solution: Since space is the set of all points, it would not make sense to consider three spaces as separate geometric figures. Realistically, there is only one space. Therefore, at least 4 points must exist. Space contains at least 4 different, non-coplanar points, no three of which are collinear. (gain, the reasoning is similar to coincident lines or planes. 173 Unit II Fundamental Terms
Unit II Fundamental Terms Part Postulates (or xioms Lesson 3 Postulate 2: Uniqueness of Lines, Planes, and Spaces bjective: To understand the assumptions regarding the uniqueness of lines, planes, and spaces, in our Geometry. Important Terms: Postulate 2 - Uniqueness of Lines, Planes, and Spaces 1st ssumption: There is exactly one line through two different points. (Stated as a conditional: If there are two different points, then there exists exactly one line containing those two points. 2nd ssumption: There is exactly one plane through three different, noncollinear points. (Stated as a conditional: If there are three different, non-collinear points, then there exists exactly one plane containing those three points. 3rd ssumption: There is exactly one space through four different, noncoplanar points, no three of which are collinear. (Stated as a conditional: If there are four different, non-coplanar points, no three of which are collinear, then there exists exactly one space containing those four points. Example 1: onsider three points,, and. How many different lines are determined if: a The points are collinear? b The points are arranged so that they form a triangle? Solution: a Exactly one line is determined. onsidering the points two at a time, we could use three different pairs of points to determine a line, but they would all be the same line., and are all the same line. b Three different lines are determined, or are all different lines. 175 Unit II Fundamental Terms
Example 2: onsider five points U, V, W, X, and Y. How could the five points be arranged so that exactly ten lines would be determined and five planes would be determined? Solution: Draw a pyramid with a quadrilateral base. Y U X V W There is exactly one line through any two different points: UV, VW, WX, XU, UY, VY, XY, UW, VX, WY There is exactly one plane through three different, non-collinear points: plane UVY, plane UVW, plane VWY, plane WXY, plane UYX Lesson 3 Exercises: 1. If we have two different points, then, we have exactly containing those. 2. If we have three different points, then we have exactly plane containing those. 3. If we have four different points, no three of which are, we have exactly space containing those. 4. onsider four points U, V, W, and X. How many different lines can be determined by these points if: a The points are collinear? b The points are arranged so that three are on the same line, and the fourth is not on that line. c The points are arranged so that no three are collinear? d The points are arranged to form a pyramid? 5. Using the same four points U, V, W, and X in Exercise 4, state how many different planes can be determined if the situations are the same as in parts a through d. Part Postulates (or xioms 176
9. Mcormick Place in hicago is one of the largest convention-expo centers in the world. It is made up of several huge rooms which could easily enclose a football field. To meet the requirements of conventions of all sizes, however, these rooms can be divided into smaller rooms by the use of floor-to-ceiling folding walls. a Suppose you use one folding wall to divide a larger room into two smaller rooms. Describe the relationship between the two smaller rooms to the larger room, (in terms of the Plane Separation ssumption. b Room and Room have been created by dividing Room with a folding wall. Describe (from a geometric viewpoint how you know a speaker in Room is not in the same half of Room as a speaker in Room. c Two exhibits are in opposite corners of Room, as described in 9b. an we say the two exhibits are in the same room (half-space? Why or why not? 10. point separates a line into two half-lines. Describe two situations in the real world that would model this geometric concept. 11. line separates a plane into two half-planes. Describe two situations in the real world that would model this geometric concept. 12. plane separates space into two half-spaces. Describe two situations in the real world that would model this geometric concept. 13. point divides a line into two half-lines. We can then say that three sets of points have been determined the points in the first half-line, the points in the second half-line, and the separation point. Describe in a similar fashion the three sets of points determined by the following: a line separating a plane into two half-planes. b plane separating space into two half-spaces. 14. If a plane separates space into two half-spaces, is each point of the plane between some point in one half-space and some point in the other half-space? Explain your answer. Extending your mind. 15. Into how many different subsets does a circle separate the points in a plane? Draw a circle, and label those sets. Use this concept to answer the questions below. a Draw a circle and select segments between points outside the circle, to show intersections of 0, 1, and 2 points with the circle itself. If any of these situations cannot exist, say so. b Draw a circle and select segments between points inside the circle, to show intersections of 0, 1, and 2 points with the circle itself. If any of these situations cannot exist, say so. c Draw a circle and select segments between points on the circle, to show intersections of 0, 1, and 2 points with the circle itself. If any of these situations cannot exist, say so. 183 Unit II Fundamental Terms
Unit II Fundamental Terms Part Postulates (or xioms Lesson 6 Postulate 5: Intersection of Lines or Planes bjective: To understand the assumptions regarding the intersection of two lines, or the intersection of two planes, in our Geometry. Important Terms: Postulate 5 - Intersection of Lines or Planes 1st ssumption: If two different lines intersect, then, the intersection is a unique point. 2nd ssumption: If two different planes intersect, then, the intersection is a unique line. Example 1: Sketch and label a prism with triangular bases. List the unique lines determined by the intersection of the faces with the bases of the prism. Solution: Plane intersects plane ED Plane intersects plane FE Plane intersects plane FD DE Plane DEF intersects plane DE EF Plane DEF intersects plane FE DF Plane DEF intersects plane FD D E F Example 2: In the figure shown in Example 1, the vertical edges determine unique points when they intersect with the triangular base DEF. List the points and the lines which determine them. Solution: D DE intersects DF E DE intersects EF F EF intersects FD Part Postulates (or xioms 184
For Exercises 4 through 8, use the given figure. X R P Y Q k l m P W Z 4. Point R is the unique intersection point of line and line. 5. Point P is the unique intersection point of line and line ; line and line ; or line and line. 6. Point Q is the unique intersection point of line and line ; line and line ; or line and line. 7. What relationship appears to exist between XW and YZ? Is there a unique intersection point for these two lines? Why or why not? 8. What relationship appears to exist between XW and line m? Must this relationship have a unique intersection point? Explain. For Exercises 9 and 10, refer to the given figure. Think of each segment in the figure as part of a line. D 9. Four unique vertical lines are determined by the planes in this diagram. Name these lines and, for each, name the planes that intersect to determine them. E H F G 10. How many unique points are determined by the lines in this diagram? Name the points which are nearest the front of the figure and the lines which determine them? 11. onsider a pyramid with a pentagonal base. How many lines are created by the intersection of the planes (or faces of the pyramid? e sure to consider the plane of the base in your thinking. 12. For the pyramid referred to in Exercise 11, how many points are determined by the lines (edges of the pyramid? Part Postulates (or xioms 186
Unit II Fundamental Terms Part Postulates (or xioms Lesson 7 Postulate 6: Ruler bjective: To understand the assumptions regarding the measuring of line segments, as they represent the distance between points, in our Geometry. Important Terms: Line Segment s the word segment comes from a Latin word meaning, to cut, this is a set of points created by cutting a line in two places, creating a piece of a line. Formally, in our Geometry, the set of points, designated XY, is a line segment, if and only if, it is the union of two points on a line, called the endpoints of the segment, and the set of all the points between them. See the illustration below. can be represented by Length of a Line Segment Formally, in our Geometry, a given real number x is the length of a line segment MN, if and only if, that real number represents the distance between the endpoints M and N, of the line segment. That length is designated by writing a lower case m in front of the symbol for the line segment, or simply by writing the symbol for the line segment without the segment bar over it, and reading it as the measure of line segment MN, or the measure of MN. See the illustration below. Postulate 6 - Ruler X Y mmn = MN = x 1st ssumption: If you have the set of all points on a line, then those points can be put into a one-to-one correspondence with all of the real numbers, in an ordered way, such that, any point may correspond to zero, and any other point may correspond to one. 2nd ssumption: If you have a pair of points on a line, then there corresponds to that pair of points, exactly one number, called the unique distance between the points. 3rd ssumption: If you have two points on a line, for which coordinates have been assigned, then the distance between those two points, is the absolute value of the difference, between their coordinates. XY 187 Unit II Fundamental Terms
Lesson 7 Exercises: 1. Using a ruler, measure the length of each of the following segments, to the nearest millimeter. a b c d e f 2. Draw a sketch of the three collinear points given in each of the following. Then write the Segment ddition ssumption specifically for those points. a E is between D and F. b H is between G and J. c M is between N and P. d R is between Q and S. 3. Using the given diagram to the right, how can you: a Determine RT if you know RS and ST? b Determine QR if you know QT and RT? Q R S T 191 Unit II Fundamental Terms
4. Which of the statements below are true and which are false? a 5 = 5 g b -17 = -17 h c -8 = 8 i d 6 = 4 + -2 j e -3 > -7 k f -14 < -14 l 12 > 12 17-6 < 17 6-15 < 0-7 - -4 < (-7 ( 4 For every x < 3, x > 2 For every x < 3, x > -3 5. opy the following number line and label each point whose coordinate is given below. -6-5 -4-3 -2-1 0 1 2 3 4 5 6 1 a 5 b -3 c d.75 e -2.25 f 2 2 6. Find the distance between each of the following pairs of points with the given coordinates. a 2 and 9 b -8 and 13 c 3 and 2 d 2.28 and 5.16 1 e f -9 and -2.4 4 and 2 3 g 14.7 and 9.2 h 3.2 and -7.9 i -2.94 and -15.17 j 5 1 4 and - 3 5 8 k x and y 7. Using the given number line, find the distance between each indicated pair of points. D E F -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 a and F b and c and F d and e E and F f and D g and D h and i and F Part Postulates (or xioms 192
4th ssumption: If, in a half-plane, a ray lies between rays and, then the measure of angle (indicated by m, plus, the measure of angle (indicated by m, is equal to the measure of angle (indicated by m. This assumption is also called the ngle ddition ssumption, and can be represented mathematically as: m + m = m Note: The Protractor Postulate can be summarized briefly, as follows: To measure an angle, you must attach numbers to the rays, and find the unique measure of that angle, by taking the absolute value of the difference, between the two coordinates. Example 1: Using the given protractor, and assuming a correspondence between rays having a common endpoint with real numbers on the arc of the protractor, find the measure of each angle named below. 80 90 100 70 110 N 60 100 80 120 110 70 50 120 60 130 40 130 50 140 140 40 30 150 30 150 20 10 160 170 20 10 160 170 0 180 0 180 a MP b QP c NP d NQ Solution: a m MP = 110 b m QP = 70 c m NP = 130 70 d m NQ = 130 = 60 Example 2: If P is between N and Q, then m NP + m PQ =. Solution: m NQ; ngle ddition ssumption. M P Q Lesson 8 Exercises: 1. Using the given protractor and assuming a correspondence between rays having a common end point with real numbers on the arc of the protractor, find the measure of each angle named below. 70 80 90 100 110 60 100 80 120 110 70 50 120 60 130 40 130 50 140 140 40 30 150 30 150 20 10 160 170 20 10 160 170 0 180 0 180 a ED b c d E e D f D D E 195 Unit II Fundamental Terms
2. Use a protractor to determine the measure of each angle named below. D E F a D b F c D d E e E f g EF h D 3. In general, if it is known that m D is 85, and m D is 35, is it then known that m is 120? 4. If m WXY = 84, m YXZ = 26, and m WXZ = 110, what do you know about the relationship between XW, XY, and XZ? 5. If N is between M and P, then a If m MN = 23, and m NP = 46, then m MP =. b If m MN = 7, and m MP = 62, then m NP =. c If m NP = 27, and m MP = 119, then m MN =. 6. Without using a protractor, estimate the measure of each of the following angles, within 10 degrees. Then use your protractor to check the accuracy of your answer. a b c d e f Part Postulates (or xioms 196
7. ompare the Protractor Postulate and the Ruler Postulate. How are they alike? How are they different? 8. ompare the ngle ddition ssumption and the Segment ddition ssumption. How are they alike? How are they different? 9. Point lies inside. Find m if: a m = 70 and m = 80. 1 b m = 24 and m = 101. 2 1 4 10.Point lies inside. Find m if: a m = 110 and m = 70. 3 b m = 130 and m = 75. 4 1 2 Use the figure to the right to answer Exercises 11 through 14. Write and solve an equation to find the measure of each angle. 11.m E = 45 and m DE = 26. Find m D. 12.m E = 80 and m D = 37. Find m DE. 13.m E = 4y, m DE = 2y, and m D = 24 Find m DE and m E. D E 14.m D = (2x +7, m DE = (3x 11, and m E = 76. Find m D and m DE. 197 Unit II Fundamental Terms
Unit II Fundamental Terms Part Postulates (or xioms Lesson 9 Postulate 8: ircle bjective: To understand the assumptions regarding the measuring of arcs of a circle, as they represent the distance between points on a circle, in our Geometry. Important Terms: rc of a ircle From a Latin word meaning, bow, this is any set of continuous points on a circle. See the illustration below. M N MN is an arc of circle Minor rc of a ircle n arc whose points are in the interior of a central angle of a circle. Formally, in our Geometry, an arc of a circle is a minor arc, if and only if, it is the union of two points on the circle (not the endpoints of a diameter, and the set of points on the circle, which lie in the interior of the angle formed by the radii containing the two points. Note: Remember that the interior of an angle is defined to be the set of points, in a half-plane, that are between the two rays of the angle. See the illustration below. is a minor arc of circle Major rc of a ircle n arc whose points are in the exterior of a central angle of a circle. Formally, in our Geometry, an arc of a circle is a major arc, if and only if, it is the union of two points on the circle (not the endpoints of a diameter, and the set of points on the circle, which lie in the exterior of the angle formed by the radii containing the two points. Note: Remember that a third point on the arc must be used to name a major arc. See the illustration below. Postulate 8 - ircle D D is a major arc of circle 1st ssumption: If you have the set of all points on a circle, then those points can be put into a one-to-one correspondence with the real numbers from 0 to 360, in an orderly fashion, with the exception of any one point, which will be paired with both 0 and 360. Part Postulates (or xioms 198
2nd ssumption: If you have two points on a circle, not the endpoints of a diameter, then there correspond to those points, exactly two real numbers whose sum is 360, each of which may be called, the distance between the two points. 3rd ssumption: If you have any two points on a circle, then the two distances between the points, as defined by the resulting two arcs, are, #1 - the absolute value of the difference between their coordinates, and, #2-360 minus that absolute value. 4th ssumption: If, on a circle, a point lies between points and, then the measure of arc (indicated by m, plus the measure of arc (indicated by m is equal to, the measure of arc (indicated by m. This assumption is also called the rc ddition ssumption, and can be represented mathematically as: m + m = m. Example 1: Using the given circle, find the distance between the indicated pairs of points. Give two distance measurements in each case. a to E 120 90 b to G c E to d to G 180 D E 240 F 270 G 0 360 330 Solution: a to E Minor rc: 90 240 Major rc: -150 150 b to G Minor rc: 360 330 Major rc: 30 30 c E to Minor rc: 240 120 Major rc: 120 120 d to G Major rc: 90 330 Minor rc: -240 240 360 150 210 360 30 330 360 120 240 360 240 120 199 Unit II Fundamental Terms
In Exercises 2 through 4, find the measure of the indicated arcs. E D 2. 3. 4. 170 G 75 70 m = m D = m EGF = m = m F 85 130 In Exercises 5 through 8, find the value of x. Then find the measure of the indicated arcs. x 5. 6. x (2x - 30 4x x D x = m = x = m = 7. 8. D 4x 6x (5x+10 7x 7x x = m D = m = (x+100 (4x+30 x = m = m = m = 9. Points P, Q, and R lie on the same circle. If = m PR + m PQR, and = 360, then tell whether >, >, =, or the relationship cannot be determined. 10.In the picture, if m = 72, m = 48, m D = 60, m DE = 72, m EF = 56, and m F = 52, which of the line segments (if any D, E, and F are diameters of the circle? Note: Do not assume any measures by appearances. 201 Unit II Fundamental Terms F D E
Example 1: (continued Solution: 1. True 8. True 2. True 9. True 3. True 10. Infinitely many 4. True 11. Yes 5. True 12. Yes (at least one 6. False 13. No 7. True 14. If you have a line in a plane and a point in the plane not on the given line, then there is one and only one line through the point which is parallel to the given line. (In other words, that line exists, and is unique. Example 2: lgebra onnection (VideoText lgebra, Unit III, Part, Lesson 2 Write the equation of the line that passes through ( 2, 5 and is parallel to 2x y = 6. Solution: a Is ( 2, 5 on 2x y = 6? 22 ( 5= 6 4 5= 6-1 6 Since the ordered pair does not satisfy the equation of the line, the point is not on the line. b Then, to find the line which passes through the point, and is parallel to the given line, we must find the slope of the given line. (The two lines must have the same slope to be parallel. That means we must rewrite the equation for the given line in the slope-intercept form, or, y = mx + b. 2x y = 6 2x y 2x = 6 2x y = 2x+ 6 ( 1 ( y= ( 1 ( 2x+ 6 y = 2x 6 This tells us that our new line through ( 2, 5 must have a slope equal 2 or. c Substituting 2 for m, we can now find the y-intercept, b, and the equation of the desired line, as follows: y = m x+ b Therefore, y = 2x+ 1 or y = 2x+ b 2x+ y = 1 or 5= 2( 2 + b 2x y = 1 5= 4+ b 5 4= 4 4+ b 1= b 2 1 Part Postulates (or xioms 204
Use the figure to the right for Exercises 4 through 6. Think of each segment in the figure as part of a line. Q R 4. Name all lines which are parallel to XW. S T 5. Name all lines which are parallel to plane UVWX. U V 6. Name all lines which are parallel to XW and pass through point R. X W In Exercises 7 through 9, tell whether each statement is true or false. 7. Exactly one line may be parallel to another line. 8. Parallel lines may exist on a line. 9. given rectangle exists in a plane, and is unique. 10.onsider the equation 2x + 3y = 12. Is the point ( 4, 7 on the solution set line for this equation? Why or why not? 11.Find the equation whose solution set line would pass through the point ( 4, 7 and would also be parallel to the solution set line for the equation 2x + 3y = 12. 12.Graph the results of Exercises 10 and 11. 13. For Exercises 10, 11, and 12, represents the line through the point which would be parallel to the line represented by. ordered pair equation equation Part Postulates (or xioms 206
Example 3: lgebra onnection (VideoText lgebra - Unit III, Part, Lesson 2 Write the equation whose solution set line passes through ( 1, 6 and is perpendicular to the solution set line for the equation 2x + 5y = 15. Solution: First, we must rewrite the equation in the slope-intercept form. 2x+ 5y = 15 5y = 2x+ 15 2 y = x+ 3 5 2 Slope is y - intercept is 0, 3 5 ( ur new line through ( 1, 6 must have a slope whose product with the slope of the given line is 1. 2 m = 1 5 5 2 5 m = ( 1( 2 5 2 5 1 m = 2 Now we must substitute the desired slope in the general slope-intercept form to find the y-intercept b. y = m x+ b 5 y = x+ b 2 5 6 = ( ( 2 1 + b 5 6 = + b 2 6 5 5 5 = + b 2 2 2 12 5 = 0 + b 2 2 7 = b 2 So the equation for the solution set line we are looking for is: y = m x+ b 5 7 y = x+ or 2 2 2 y = 5x + 7 or 5 x + 2 y = 7 or 5 x 2y = 7 209 Unit II Fundamental Terms
In Exercises 10 through 12, trace the given figure, sketch the apothem of the polygon (Unit I WorkText, page 58, name the necessary points appropriately, identify the line containing the line segment representing the apothem, and identify the line containing the line segment that is the side to which the apothem is drawn. (answers will vary depending on the points you name S 10. 11. T 12. R U Y V D 13.Sketch a right pyramid of your choice. n one face, draw the slant height (Unit I WorkText, page 71. Name the necessary points appropriately, identify the line containing the line segment representing the slant height, and identify the line containing the line segment that is the base to which the slant height is drawn. X W F G H 14. Restate, in writing, the portion of Postulate 10 which allows each altitude, apothem, and slant height in Exercises 3 through 13 to be drawn. 15.Sketch a right pyramid of your choice. Draw the altitude of the pyramid (Unit I WorkText, page 71. Name the necessary points appropriately, identify the line containing the line segment representing the altitude of the pyramid, and identify the plane containing the base to which the altitude is drawn. 16. Restate, in writing, the portion of Postulate 10 which allows the altitude in Exercise 15 to be drawn. 17.n a piece of graph paper, sketch the solution set line for the equation of and plot the point ( -3, -5. a Show algebraically that the given point is not on the given line. b Find the equation for the solution set line through ( -3, -5 which is also perpendicular to the solution set line for the equation y = 3 x + 4. 2 c Sketch the graph of the perpendicular line found in part b. y y = 3 x + 4 2 x Part Postulates (or xioms 212