Teacher Annotated Edition. Study Notebook

Transcription

1 Teacher Annotated Edition Study Notebook

2 Copyright by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without prior written permission of the publisher. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH ISBN: MHID: Printed in the United States of America

3 Contents Chapter 1 Before You Read... 1 Key Points Points, Lines, and Planes Linear Measure and Precision Distance and Midpoints Angle Measure Angle Relationships Two-Dimensional Figures Three-Dimensional Figures Tie It Together Before the Test Chapter 2 Before You Read Key Points Inductive Reasoning and Conjecture Logic Conditional Statements Deductive Reasoning Postulates and Paragraph Proofs Algebraic Proof Proving Segment Relationships Proving Angle Relationships Tie It Together Before the Test Chapter 3 Before You Read Key Points Parallel Lines and Transversals Angles and Parallel Lines Slopes of Lines Equations of Lines Proving Lines Parallel Perpendiculars and Distance Tie It Together Before the Test Chapter 4 Before You Read Key Points Classifying Triangles Angles of Triangles Congruent Triangles Proving Congruence: SSS, SAS Proving Congruence: ASA, AAS Isosceles and Equilateral Triangles Congruence Transformations Triangles and Coordinate Proof Tie It Together Before the Test Chapter 5 Before You Read Key Points Bisectors of Triangles Medians and Altitudes of Triangles Inequalities in One Triangle Indirect Proof The Triangle Inequality Inequalities in Two Triangles Tie It Together Before the Test Chapter 6 Before You Read Key Points Angles of Polygons Parallelograms Tests for Parallelograms Rectangles Rhombi and Squares Kites and Trapezoids Tie It Together Before the Test Chapter 7 Before You Read Key Points Ratios and Proportions Similar Polygons Similar Triangles Parallel Lines and Proportional Parts Parts of Similar Triangles Similarity Transformations Scale Drawings and Models Tie It Together Before the Test Chapter 8 Before You Read Key Points Geometric Mean The Pythagorean Theorem and Its Converse Special Right Triangles Trigonometry Angles of Elevation and Depression The Law of Sines and Cosines Vectors Tie It Together Before the Test iii

4 Chapter 9 Before You Read Key Points Refl ections Translations Rotations Compositions of Transformations Symmetry Dilations Tie It Together Before the Test Chapter 10 Before You Read Key Points Circles and Circumference Measuring Angles and Arcs Arcs and Chords Inscribed Angles Tangents Secants, Tangents, and Angle Measures Special Segments in a Circle Equations of Circles Tie It Together Before the Test Chapter 11 Before You Read Key Points Areas of Parallelograms and Triangles Areas of Trapezoids, Rhombi and Kites Areas of Circles and Sectors Areas of Regular Polygons and Composite Figures Areas of Similar Figures Tie It Together Before the Test Chapter 12 Before You Read Key Points Representations of Three-Dimensional Figures Surface Areas of Prisms and Cylinders Surface Areas of Pyramids and Cones Volumes of Prisms and Cylinders Volumes of Pyramids and Cones Surface Areas and Volumes of Spheres Spherical Geometry Congruent and Similar Solids Tie It Together Before the Test Chapter 13 Before You Read Key Points Representing Sample Spaces Permutations and Combinations Geometric Probability Simulations Probabilities of Independent and Dependent Events Probabilities of Mutually Exclusive Events Tie It Together Before the Test iv

5 Note-Taking Tips Your notes are a reminder of what you learned in class. Taking good notes can help you succeed in mathematics. The following tips will help you take better classroom notes. Before class, ask what your teacher will be discussing in class. Review mentally what you already know about the concept. Be an active listener. Focus on what your teacher is saying. Listen for important concepts. Pay attention to words, examples, and/or diagrams your teacher emphasizes. Write your notes as clear and concise as possible. The following symbols and abbreviations may be helpful in your note-taking. Word or Phrase Symbol or Abbreviation Word or Phrase Symbol or Abbreviation for example e.g. not equal such as i.e. approximately with w/ therefore without w/o versus vs and + angle Use a symbol such as a star ( ) or an asterisk ( ) to emphasis important concepts. Place a question mark (?) next to anything that you do not understand. Ask questions and participate in class discussion. Draw and label pictures or diagrams to help clarify a concept. When working out an example, write what you are doing to solve the problem next to each step. Be sure to use your own words. Review your notes as soon as possible after class. During this time, organize and summarize new concepts and clarify misunderstandings. Note-Taking Don ts Don t write every word. Concentrate on the main ideas and concepts. Don t use someone else s notes as they may not make sense. Don t doodle. It distracts you from listening actively. Don t lose focus or you will become lost in your note-taking. v

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7 The Tools of Geometry Before You Read Before you read the chapter, respond to these statements. 1. Write an A if you agree with the statement. 2. Write a D if you disagree with the statement. Before You Read See students work. The Tools of Geometry Collinear points are lines that run through the same point. Line segments that are congruent have the same measure. Distance on a coordinate plane is calculated using a form of the Pythagorean Theorem. Note Taking Tips The Midpoint Formula can be used to find the coordinates of the endpoint of a segment. The formula to find the area of a circle is A = πr 2. Study Organizer Construct the Foldable as directed at the beginning of this chapter. When taking notes, summarize the main ideas presented in the lesson. Summaries are useful for condensing data and realizing what is important. When you take notes, write descriptive paragraphs about your learning experiences. Chapter 1 1 Glencoe Geometry

8 The Tools of Geometry Key Points Scan the pages in the chapter and write at least one specific fact concerning each lesson. For example, in the lesson on linear measure, one fact might be that unlike a line, a line segment can be measured because it has two endpoints. After completing the chapter, you can use this table to review for your chapter test. Lesson Fact 1-1 Points, Lines, and Planes See students work. 1-2 Linear Measure 1-3 Distance and Midpoints 1-4 Angle Measure 1-5 Angle Relationship 1-6 Two-Dimensional Figures 1-7 Three-Dimensional Figures Chapter 1 2 Glencoe Geometry

9 1-1 Points, Lines, and Planes What You ll Learn Scan the text in Lesson 1-1. Write two facts you learned about points, lines, and planes as you scanned the text. Sample answer: A line does not have thickness or 1. width. Active Vocabulary Sample answer: Collinear points are points on the 2. same line. New Vocabulary Write the definition next to each term. Lesson 1-1 undefined term point line can only be explained using examples and descriptions a location that has neither shape nor size made up of points and has no thickness or width plane collinear coplanar intersection definition/defined term flat surface made up of points that extends indefinitely in all directions points that lie on the same line points that lie on the same plane the set of points that two or more geometric figures have in common can be explained using undefined terms and/or other defined terms space a boundless, three-dimensional set of all points Chapter 1 3 Glencoe Geometry

10 Lesson 1-1 (continued) Main Idea Details Points, Lines, and Planes pp. 5 6 Model a point, line, and plane with a representative drawing. Label your drawing. Sample answers: point line plane Intersections of Lines and Planes pp. 6 7 Compare undefined and defined terms by completing the table. Provide definitions and examples. Term Definition Examples Helping You Remember defined term undefined term explained using undefined terms or other defined terms; same as definition can only be explained using examples and descriptions space, intersection, collinear, coplanar, intersection points, lines, planes Recall or look in a dictionary for the meaning of the prefix co-. What does the prefix mean? How can it help you remember the meaning of collinear? Sample answer: joint or working together; two or more points together Chapter 1 4 Glencoe Geometry

11 1-2 Linear Measure What You ll Learn Scan the text under the Now heading. List two things you will learn about in this lesson. 1. Sample answer: how to measure segments 2. Sample answer: how to calculate with measures Active Vocabulary New Vocabulary Fill in each blank with the correct term or phrase. betweenness of points line segment Between any two points on a line there is another point. a portion of a line endpoints with two Lesson 1-2 between construction congruent segment point A is between two other points, B and C; A, B, and C are on the same line, and AB + AC = BC method of creating measuring tools two segments geometric figures with the same measures without Vocabulary Link Congruent segments can be illustrated by real-world examples. Consider opposite sides of this book. The lines are the same size. Write some real-world examples of other congruent segments. Sample answer: railroad tracks, edges of picture frame Chapter 1 5 Glencoe Geometry

12 Lesson 1-2 (continued) Main Idea Details Measure Line Segments pp Use the model to fill in each blanks. A B 1. Each inch is divided into. eighths Point B is closer to the 8 inch mark AB is about 8 inches long. Calculate Measures pp Model the three types of methods to calculate measures of a line segment with an example. Helping You Remember Find measurements by adding. Sample answer: number line with two segments that are equal to a longer segment Find measurements by subtracting. Sample answer: number line with a smaller segment subtracted from a larger segment Find measurements by writing and solving equations. Sample answer: ST = 4x, SV = 5x - 3, TV = 15 Find the value of x and ST. 4x + 5x - 3 = 15 x = 2; ST = 8 Construction is a word used in everyday English as well as in mathematics. Look up construction in the dictionary. Explain how the everyday definition can help you remember how construction is used in mathematics. Sample answer: The way in which something is put together; figures are put together just like buildings and structures. Chapter 1 6 Glencoe Geometry

13 1-3 Distance and Midpoints What You ll Learn Skim Lesson 1-3. Predict two things that you expect to learn based on the headings and the Key Concept box. Sample answer: how to find the distance between 1. two points Sample answer: how to find the midpoint between 2. two points Active Vocabulary New Vocabulary Write the definition next to each term. distance the length of a segment with two endpoints midpoint point that is halfway between the endpoints segment bisector segment, line, plane that intersects a segment at its midpoint Vocabulary Link Midpoint can have non-mathematical meanings as well. Look up midpoint in the dictionary. Explain how the English definition can help you remember how midpoint is used in mathematics. Sample answer: a point in time halfway between the beginning and end; a point halfway between Lesson 1-3 two extremes Chapter 1 7 Glencoe Geometry

14 Lesson 1-3 (continued) Main Idea Details Distance Between Two Points pp Fill in the organizer for the distance formula on a number line and a coordinate plane with line segment AB, with endpoints (x 1, y 1 ) and (x 2, y 2 ). Distance Formula Number Line Coordinate Plane AB = x 1 - x 2 or x 2 - x 1 Symbols AB = (x 2 - x 1 ) 2 + (y 2 - y 1 ) 2 Midpoint of a Segment pp The distance between two points is the absolute value of the difference between their coordinates. Model the solution to find the distance between the points on the coordinate plane. Use the lines to show your calculations. Round to the nearest tenth if necessary. (x 1, y 1 ) = (-5, 3) ; ( x 2, y 2 ) = ( 2, 4 ) KL = (x 2 - x 1 ) 2 + (y 2 - y 1 ) 2 KL = (2 - (-5)) 2 + (4-3) 2 KL = Words If a segment has endpoints with coordinates (x 1, y 1 ) and (x 2, y 2 ), you can find the distance with the distance formula. y x KL = Chapter 1 8 Glencoe Geometry

15 1-4 Angle Measure What You ll Learn Skim the Examples for Lesson 1-4. Predict two things you think you will learn about angle measures. Sample answer: how to measure angles 1. Sample answer: how to classify angles 2. Active Vocabulary New Vocabulary Write the correct term next to each definition. right angle angle with measure of 90º vertex where two rays meet to form an angle obtuse angle angle with measure greater than 90º angle bisector a ray that divides an angle into two congruent angles interior acute angle side ray angle degree opposite rays the region of a plane inside of an angle angle with measure less than 90º a ray that forms part of an angle part of a line with one endpoint and one end that extends forever formed by two noncollinear rays with a common endpoint units used to measure angles extend in two directions from a point on a line Lesson 1-4 exterior the region of a plane outside of an angle Chapter 1 9 Glencoe Geometry

16 Lesson 1-4 (continued) Main Idea Measure and Classify Angles pp Details Summarize information about angles in the graphic organizer below. 1. Draw and label an obtuse, acute, and right angle. acute obtuse right 2. Draw and label one angle that has these 3 names: B, ABC, and 3. 3 Congruent Angles pp Model a pair of congruent angles. Use a compass and straightedge. Drawings will vary, but both angles have the same measure. Sample answer shown. Helping You Remember congruent angles. Compare and contrast congruent segments and Sample answer: Congruent segments and congruent angles have the same measure. Segments have a linear measure. Angles are measured in degrees. Chapter 1 10 Glencoe Geometry

17 1-5 Angle Relationships What You ll Learn Scan Lesson 1-5. List two headings you would use to make an outline of this lesson. Sample answer: Special Pairs of Lines 1. Sample answer: Perpendicular Lines 2. Active Vocabulary New Vocabulary Label the diagram with the correct terms. Use each term once. adjacent linear pair vertical complementary supplementary perpendicular 1 and 2 are supplementary angles and a linear pair. w Lines w and x are perpendicular x Vocabulary Link Perpendicular has a nonmathematical meaning. Look up perpendicular in the dictionary. List at least five examples of things that are perpendicular in everyday life. Sample answer: lines of corner of book, desk, room, refrigerator and floor, wall and ceiling 4 4 and 5 are complementary angles and adjacent angles. 1 and 3 are vertical angles. Lesson 1-5 Chapter 1 11 Glencoe Geometry

18 Lesson 1-5 (continued) Main Idea Details Pairs of Angles pp Summarize angle relationships by completing the graphic organizer with the terms adjacent angles, complementary angles, vertical angles, and supplementary angles. adjacent angles relationship due to positioning vertical angles Special angle pairs relationship due to angle measures complementary angles supplementary angles Perpendicular Lines pp Helping You Remember If m ABC = 2x + 12, find x so that BC BA. 2x + 12 = 90 definition of perpendicular x = 39 Simplify. Look up the nonmathematical meaning of supplementary in a dictionary. How can this definition help you to remember the meaning of supplementary angles? Sample answer: something added to complete a deficiency; two angles added together to be a complete line Chapter 1 12 Glencoe Geometry

19 1-6 Two-Dimensional Figures What You ll Learn Skim the lesson. Write two things you already know about two-dimensional figures. Sample answer: Polygons are two-dimensional 1. figures. Active Vocabulary Sample answer: Perimeter and area of 2. two-dimensional figures can be found. New Vocabulary Write each term next to its definition. Lesson 1-6 vertex of polygon where two sides of a polygon intersect convex polygon with no side that passes through the figure s interior regular polygon a convex polygon that is equilateral and equiangular equiangular polygon circumference n-gon polygon concave equilateral polygon perimeter a polygon where all angles are congruent perimeter of a circle a polygon with n number of sides closed figure formed by coplanar segments where two segments intersect at a common noncollinear endpoint polygon with some side that passes through the figure s interior a polygon where all sides are congruent sum of the lengths of a polygon s sides area number of square units needed to cover a surface Chapter 1 13 Glencoe Geometry

20 Lesson 1-6 (continued) Main Idea Details Identify Polygons pp Fill in the organizer about polygons. What is a polygon? noncollinear vertices closed figure made of coplanar segments What is not a polygon? open curved sides intersecting segments Examples Polygons Nonexamples Perimeter, Circumference, and Area pp Summarize the formulas for perimeter, circumference, and area of the figures by completing the chart. Area Perimeter A = 1 2 bh A = s2 A = l w πr 2 P = a + b + c P = 4s P = 2l + 2w Circumference n/a n/a n/a same as circumference C =πd or C = 2πr Chapter 1 14 Glencoe Geometry

21 1-7 Three-Dimensional Figures What You ll Learn Scan the text in Lesson 1-7. Write two facts you learned about three-dimensional figures. Sample answer: A polyhedron is a solid with all 1. flat surfaces that enclose a single region or space. Sample answer: There are exactly five types of 2. regular polyhedron. Active Vocabulary prism New Vocabulary Write the definition next to each term. polyhedron with two bases connected by Lesson 1-7 parallelogram faces pyramid cylinder base cone polyhedron with a polygonal base and three or more triangular faces that meet at a common vertex solid with congruent parallel circular bases and curved side parallel, congruent faces that intersect faces solid with a circular base connected by a curved surface to a single vertex sphere set of points equidistant from a given point Chapter 1 15 Glencoe Geometry

22 Lesson 1-7 (continued) Main Idea Details Identify Three-Dimensional Figures pp Complete the concept circle by drawing in a fourth example. Then state the relationship of the 4 figures on the line below the concept circle. Drawings will vary; must be a prism. The relationship is: All are prisms. Surface Area and Volume pp Figure Surface Area Helping You Remember Organize information about volume and surface area in the chart. T = Ph + 2B T = 1 2 Pl + B T = 2πrh + 2πr2 T = πrl + πr 2 T = 4πr2 Volume V = Bh V = 1 3 bh V = πr2 h V = 1 3 πr 2 h V = 4 3 πr 3 A good way to remember the characteristics of geometric solids is to think of how different solids are alike. Name a way which pyramids and cones are alike. Sample answer: Pyramids and cones both have one common vertex in which the sides or faces are attached. Chapter 1 16 Glencoe Geometry

23 The Tools of Geometry Tie It Together Complete the graphic organizer with a term or formula from the chapter. Vertical Linear Pair Complementary Adjacent Angle Pairs Supplementary Length/Distance Formula d = (x 2 - x 1 ) + (y 2 - y 1 ) Segment Right Midpoint Formula ( (x + x ) 1 2, (y + y 1 2 ) ) 2 2 Line Point Plane Angle Obtuse Perpendicular Basic Geometry Cylinder Perimeter Area Polygons Equilateral Equiangular Volume Surface Area 3D Figures Acute Cone Sphere Polyhedron Prism Regular Pyramid Chapter 1 17 Glencoe Geometry

24 The Tools of Geometry Before the Test Now that you have read and worked through the chapter, think about what you have learned and complete the table below. Compare your previous answers with these. 1. Write an A if you agree with the statement. 2. Write a D if you disagree with the statement. The Tools of Geometry After You Read Collinear points are lines that run through the same point. Line segments that are congruent have the same measure. Distance on a coordinate plane is calculated using a form of the Pythagorean Theorem. The Midpoint Formula can be used to find the coordinates of the endpoint of a segment. The formula to find the area of a circle is A = πr 2. D A A A A Math Online Visit glencoe.com to access your textbook, more examples, self-check quizzes, personal tutors, and practice tests to help you study for concepts in Chapter 1. Are You Ready for the Chapter Test? Use this checklist to help you study. I used my Foldable to complete the review of all or most lessons. I completed the Chapter 1 Study Guide and Review in the textbook. I took the Chapter 1 Practice Test in the textbook. I used the online resources for additional review options. I reviewed my homework assignments and made corrections to incorrect problems. I reviewed all vocabulary from the chapter and their definitions. Study Tips Make a calendar that includes all of your daily classes. Besides writing down all assignments and due dates, include in your daily schedule time to study, work on projects, and reviewing notes you took during class that day. Chapter 1 18 Glencoe Geometry

25 Reasoning and Proof Before You Read Before you read the chapter, respond to these statements. 1. Write an A if you agree with the statement. 2. Write a D if you disagree with the statement. Before You Read See students work. Reasoning and Proof Listing a counterexample is a method to prove a conjecture is true. A Venn diagram can illustrate a conjunction. A form of inductive reasoning is the Law of Detachment. The statement a line contains two points, is an example of a postulate. Note Taking Tips Supplementary angles have measures whose sum is 90º. Study Organizer Construct the Foldable as directed at the beginning of this chapter. When you take notes, you may wish to use a highlighting marker to emphasize important concepts. When you take notes, think about the order in which the concepts are being presented. Write why you think the concepts were presented in this sequence. Chapter 2 19 Glencoe Geometry

26 The Reasoning and Proof Key Points Scan the pages in the chapter and write at least one specific fact concerning each lesson. For example, in the lesson on inductive reasoning and conjecture, one fact might be a concluding statement reached using inductive reasoning is called a conjecture. After completing the chapter, you can use this table to review for your chapter test. Lesson Fact 2-1 Inductive Reasoning and Conjecture See students work. 2-2 Logic 2-3 Conditional Statements 2-4 Deductive Reasoning 2-5 Postulates and Paragraph Proofs 2-6 Algebraic Proof 2-7 Proving Segment Relationships 2-8 Proving Angle Relationships Chapter 2 20 Glencoe Geometry

27 2-1 Inductive Reasoning and Conjecture What You ll Learn Scan Lesson 2-1. List two headings you would use to make an outline of this lesson. Sample answer: Making Conjectures 1. Active Vocabulary Sample answer: Finding Counterexamples 2. New Vocabulary Fill in each blank with the correct term or phrase. Lesson 2-1 inductive reasoning reasoning that uses a number of specific examples to arrive at a conclusion conjecture counterexample the conclusion that is reached within inductive reasoning an example that disproves a conjecture Vocabulary Link Conjecture is a word that is used in everyday English. Find the definition of conjecture using a dictionary. Write how the definition of conjecture can help you remember the mathematical definition of conjecture. Sample answer: an expression of an opinion; A conclusion about a statement is also an expression of an opinion. Chapter 2 21 Glencoe Geometry

28 Lesson 2-1 (continued) Main Idea Make Conjectures pp Details Sequence the steps of making a conjecture with algebraic terms or geometric terms. Look for operations or attributes that repeat. Test the conjecture. Ensure all consecutive terms follow the same repeating operation or attribute. Make a conjecture. Find Counterexamples p. 92 Write a statement in which you can make a true conjecture. Then write another statement in which you can make a false conjecture. Provide a counterexample of the false conjecture. True Conjecture: Sample answer: The product of two Helping You Remember odd numbers is an even number. False Conjecture: Sample answer: The product of a negative number and a positive number is a positive number. Sample answer: -4 5 = -20 Write a short sentence that can help you remember why it only takes one counterexample to prove that a conjecture is false. Sample answer: A conjecture must be true in all cases, so one counterexample is enough to disprove a conjecture. Chapter 2 22 Glencoe Geometry

29 2-2 Logic What You ll Learn Skim the Examples for Lesson 2-2. Predict two things you think you will learn about logic. Sample answer: how to identify the truth of a 1. statement Sample answer: how to use Venn diagrams 2. Active Vocabulary New Vocabulary Match the term with the correct definition by drawing a line between the two. truth table disjunction a sentence that is either true or false two or more statements joined by and or or Lesson 2-2 statement a compound statement that uses the word or truth value conjunction negation compound statement the value of a statement as either true or false statement with the opposite meaning and opposite truth a compound statement that uses the word and convenient method to determine the truth value of statement Vocabulary Link Negation is a word used in everyday English as well as in mathematics. Look up negation in the dictionary. Explain how the English definition can help you remember how negation is used in mathematics. Sample answer: denying; Negation denies the truth of a statement. Chapter 2 23 Glencoe Geometry

30 Lesson 2-2 (continued) Main Idea Determine Truth Values pp Details Fill in the blanks to summarize negations, conjunctions, and disjunctions. Write a description of each term. truth value Statement negation disjunction compound statement conjunction Venn Diagrams pp Model the situation using a Venn diagram. At Terrace Middle school, 68 students play basketball, 77 play volleyball, 19 play soccer and basketball, and 27 play all three sports. If 13 students play both volleyball and basketball, how many students play just basketball? 9 Basketball 9 Helping You Remember Volleyball Soccer Prefixes can often help you remember the meaning of words or distinguish between similar words. Use your dictionary to find the meanings of the prefixes con and dis and explain how these meanings can help you remember the difference between a conjunction and disjunction. Sample answer: Con means together, and dis means undo or do the opposite. Together means and and opposite means or. Chapter 2 24 Glencoe Geometry

31 2-3 Conditional Statements What You ll Learn Skim Lesson 2-3. Predict two things you will learn based on the headings and the Key Concept box. Sample answer: definition of a conditional 1. statement Sample answer: definition of hypothesis and 2. conclusion Active Vocabulary New Vocabulary Write the definition next to each term. conditional statement a statement that can be written in if-then form if-then statement in the form if p, then q converse formed by exchanging the hypothesis and conclusion of the conditional hypothesis inverse contrapositive logically equivalent related conditionals the phrase immediately following the word if in a conditional statement formed by negating both the hypothesis and conclusion of the conditional formed by negating both the hypothesis and conclusion of the converse of the conditional statements with the same truth value other statements based on a given conditional Lesson 2-3 statement conclusion the phrase immediately following the word then in a conditional statement Chapter 2 25 Glencoe Geometry

32 Lesson 2-3 (continued) Main Idea If-Then Statements pp Details Identify the hypothesis and conclusion of each conditional statement. Circle the hypothesis and underline the conclusion. 1. A number is divisible by 9 if the sum of its digits is divisible by If the measure of an angle is less than 90 degrees, it is an acute angle. Related Conditionals pp Fill in the organizer for related conditionals. Symbols: p q Words: conditional statement; if p then q Symbols: -p -q Words: inverse; if not p then not q Helping You Remember Symbols: q p Words: converse; if q then p Conditional Statements Symbols: -q -p Words: contrapositive; if not q then not p When working with a conditional statement and its three related conditional, what is an easy way to remember which statements are logically equivalent? Sample answer: The two terms that contain verse (converse and inverse) are logically equivalent. The other two terms are also logically equivalent. Chapter 2 26 Glencoe Geometry

33 2-4 Deductive Reasoning What You ll Learn Scan the text in Lesson 2-4. Write two facts you learned about deductive reasoning as you scanned the text. Sample answer: Deductive reasoning uses facts, 1. rules, definitions, or properties to reach logical conclusions. Sample answer: One valid form of deductive 2. reasoning is the Law of Detachment. Active Vocabulary New Vocabulary Write the correct term next to each definition. Law of Syllogism allows you to draw conclusions from two true conditional statements when the conclusion of one statement is the hypothesis of the other deductive reasoning Law of Detachment uses facts, rules, definitions, or properties to reach logical conclusions from given statements form of deductive reasoning that states that if all the facts are true, then the conclusion reached is also true Vocabulary Link Syllogism is a word used in everyday English as well as in mathematics. Look up syllogism in the dictionary. Explain how the English definition can help you remember how syllogism is used in mathematics. Sample answer: logic; consists of major, minor premise, and conclusion; is analogy to drawing Lesson 2-4 conclusion based on two true conditional statements Chapter 2 27 Glencoe Geometry

34 Lesson 2-4 (continued) Main Idea Law of Detachment pp Details Identity the steps to validate the two conclusions by completing the graphic organizer. If the light bulb is broken, the lamp will not work. Jackson s lamp will not work. Jackson s light bulb is broken. Identify the hypothesis. The light bulb is broken. Identify the conclusion. The lamp will not work. Analyze the conclusion. Jackson s light bulb is broken. Invalid; the lamp could be broken. Law of Syllogism pp Match the portions of each statement with the correct term by drawing a line to connect the two. Then write the true conditional statement and the valid conclusion using the Law of Syllogism. Helping You Remember p q r you buy bread you walk to the store you can make toast If you walk to the store, you can buy bread. If you buy bread, you can make toast. A good way to remember something is to explain it to someone else. Suppose that a classmate is having trouble remembering the Law of Detachment. In your own words, explain what the Law of Detachment means? Sample answer: a logical argument where if the first and second statements are true, you can conclude that a third statement connecting the two true statements is true Chapter 2 28 Glencoe Geometry

35 2-5 Postulates and Paragraph Proofs What You ll Learn Scan the text under the Now heading. List two things you will learn about in this lesson. Sample answer: using basic postulates about 1. points, lines, and planes Sample answer: writing paragraph proofs 2. Active Vocabulary New Vocabulary Fill in each blank with the correct term or phrase. postulate a statement that is accepted as true without proof proof deductive argument theorem paragraph proof informal proof axiom a logical argument in which each statement you make is supported by a statement that is accepted as true a logical chain of statements that link the given to what you are trying to prove statement a or conjecture that has been proven paragraph writing a conjecture to explain why a for a given situation is true same as paragraph proof same as postulate Lesson 2-5 Chapter 2 29 Glencoe Geometry

36 Lesson 2-5 (continued) Main Idea Points, Lines, and Planes pp Details Write a postulate of points, lines, and planes. Then model the postulate. Through any three noncollinear points, there is exactly one plane. Sample answer: Paragraph Proofs pp Sequence the steps in the proof process by completing the organizer. Steps in the Proof Process Step 1: List the given ; information draw a picture if possible. Step 2: State the theorem or conjecture to be. proven Step 3: Create a deductive argument by forming a logical chain of statements linking the given to what you are trying to prove. Step 4: Justify each statement with a reason. Reasons include definitions, algebraic, properties postulates, and. theorems Step 5: State what has been. proven Chapter 2 30 Glencoe Geometry

37 2-6 Algebraic Proof What You ll Learn Skim the lesson. Write two things you already know about algebraic proof. Sample answer: properties of real numbers 1. Active Vocabulary Sample answer: how to solve an equation 2. Review Vocabulary Write the definition next to the term. (Lesson 2-4) Lesson 2-6 deductive reasoning uses facts, rules, definitions, and properties to reach logical conclusions from given statements New Vocabulary Match the term with its definition by drawing a line to connect the term. algebraic proof two-column proof formal proof same as formal proof contains statements and reasons in two columns uses a group of algebraic steps to solve problems and justify each step. Vocabulary Link Proof is a word that is used in everyday English. Find the definition of proof using a dictionary. Explain how the English definition can help you remember how proof is used in mathematics. Sample answer: evidence sufficient enough to establish something as true; Evidence is the same as justification with postulates. Chapter 2 31 Glencoe Geometry

38 Lesson 2-6 (continued) Main Idea Algebraic Proof pp Details Write a 2-column proof by completing each step. Given: x = x Prove: x = Statements Reasons x = x Given 3 ( x ) = 3 (6-1 3 x ) Multiplication Prop of Equality 8 + 3x = 18 - x Distributive Property 8 + 3x + x = 18 - x + x Addition Prop of Equality 8 + 4x = 18 Simplify. Geometric Proof p x - 8 = x = x = 10 4 x = 10 4 or Match the property with its example by drawing a line to connect the example. Symmetric Property Reflective Property Subtraction Prop of Equality Substitution Prop of Equality Division Prop of Equality Substitution Prop of Equality If ST = WX, and WX = YZ, then ST = YZ. If C = D, then D = C. Transitive Property A = A Chapter 2 32 Glencoe Geometry

39 2-7 Proving Segment Relationships What You ll Learn Scan the text in Lesson 2-7. Write two facts you learned about proving segment relationships. Sample answer: You can use a ruler or segment 1. addition to show congruent segments. Sample answer: There are postulates about 2. congruence segments. Active Vocabulary Review Vocabulary Write the definition next to each term. (Lessons 2-1 and 2-3) inductive reasoning conjecture reasoning that uses a number of specific examples to arrive at a conclusion the conclusion that is reached within inductive Lesson 2-7 counterexample conditional statement converse contrapositive inverse reasoning an example that disproves a conjecture a statement that can be written in if-then form formed by exchanging the hypothesis and conclusion of the conditional formed by negating both the hypothesis and conclusion of the converse of the conditional formed by negating both the hypothesis and conclusion of the conditional Chapter 2 33 Glencoe Geometry

40 Lesson 2-7 (continued) Main Idea Ruler Postulate pp Details Use the Model and Fill in each blank to summarize the Segment Addition Postulate. If P, Q, and R are, collinear then point Q is between P and R if and only if PQ + QR = PR. Segment Congruence pp Complete the graphic organizer to prove the statement below. Given: QR ST ; ST UV Prove: QR UV QR ST and ST UV QR UV by the definition of congruence. Helping You Remember QR = ST and ST = UV by the definition of. congruence QR = UV by the Transitive. Property A good way to keep the names straight in your mind is to associate something in the name of the postulate with the content of the postulate. How can you use this idea to distinguish between the Rule Postulate and the Segment Addition Postulate? Sample answer: There are two words in Ruler Postulate, and three words in Segment Addition Postulate. Ruler Postulate has two points in the statement, and Segment Addition Postulate mentions three points. Chapter 2 34 Glencoe Geometry

41 2-8 Proving Angle Relationships What You ll Learn Scan Lesson 2-8. List two headings you would use to make an outline of this lesson. Sample answer: Supplementary and 1. Complementary Angles Sample answer: Congruent Right Angles 2. Active Vocabulary Review Vocabulary Fill in each blank with the correct term or phrase. (Lessons 1-4 and 1-5) complementary angles two adjacent angles with measures that have a sum of 90 degrees adjacent angles angles with a common side supplemetary angles interior exterior acute angle two adjacent angles with measures that have a sum of 180 degrees the region of a plane inside an angle the region of a plane outside an angle an angle with a measure of less than 90 degrees Lesson 2-8 obtuse angle an angle with a measure of greater than 90 degrees Chapter 2 35 Glencoe Geometry

42 Lesson 2-8 (continued) Main Idea Supplementary and Complementary Angles pp Details Compare and contrast the Angle Addition Postulate and the Segment Addition Postulate by completing the Venn diagram. Angle Addition Postulate Segment Addition Postulate add adjacent angles sum of parts is measure of larger figure add collinear segments Congruent Angles pp Write a two-column proof to prove the statement by completing the chart. Given: m 2 = Prove: m 1 = m 2 = Statements 2. m 3 = m 1 3. m 1 + m 3 + m 2 = Given Reasons 2. Definition of congruent 3. Definition of complementary angles 4. m 1 + m = Substitution 5. m 1 + m = Subtraction Property 6. m 1 + m 1 = Substitution 7. m 1 + m 1 2 = Division Property 8. m 1 = Substitution Chapter 2 36 Glencoe Geometry

43 Reasoning and Proof Tie It Together Complete the graphic organizer with a term from the chapter. the process of justifying algebraic steps with properties of real numbers algebraic proof the process of justifying geometric statements with postulates or theores geometric proof If p -> q is a true statement, and p is true, then q is true. Law of Detachment the process of justifying statements based on past observations inductive reasoning the process of justifying statements with previously accepted statements deductive reasoning If p -> q and q -> r, then p -> r. Law of Syllogism If hypothesis ~p negation Logic Statements If-Then Statements p -> q conditional statements p q conjunction q p disjunction Then conclusion ~p -> p inverse q -> p converse ~q -> ~p contrapositive Chapter 2 37 Glencoe Geometry

44 Reasoning and Proof Before the Test Now that you have read and worked through the chapter, think about what you have learned and complete the table below. Compare your previous answers with these. 1. Write an A if you agree with the statement. 2. Write a D if you disagree with the statement. Reasoning and Proof After You Read Listing a counterexample is a method to prove a conjecture is true. A Venn diagram can illustrate a conjunction. A form of inductive reasoning is the Law of Detachment. The statement a line contains two points, is an example of a postulate. Supplementary angles have measures whose sum is 90. D A D A D Math Online Visit glencoe.com to access your textbook, more examples, self-check quizzes, personal tutors, and practice tests to help you study for concepts in Chapter 2. Are You Ready for the Chapter Test? Use this checklist to help you study. I used my Foldable to complete the review of all or most lessons. I completed the Chapter 2 Study Guide and Review in the textbook. I took the Chapter 2 Practice Test in the textbook. I used the online resources for additional review options. I reviewed my homework assignments and made corrections to incorrect problems. I reviewed all vocabulary from the chapter and their definitions. Study Tips Make up acronyms to remember lists or sequences. PEMDAS is one acronym for remembering the order of operations (parentheses, exponents, multiply and divide, add and subtract). (Please Excuse My Dear Aunt Sally). Chapter 2 38 Glencoe Geometry

45 Parallel and Perpendicular Lines Before You Read Before you read the chapter, think about what you know about the topic. List three things you already know about parallel and perpendicular lines in the first column. Then list three things you would like to learn about them in the second column. K What I know See students work. W What I want to find out Note Taking Tips Study Organizer Construct the Foldable as directed at the beginning of this chapter. When you take notes, preview the lesson and make generalizations about what you think you will learn. Then compare that with what you actually learned after each lesson. Before each lesson, skim through the lesson and write any questions that come to mind in your notes. As you work through the lesson, record the answer to your question. Chapter 3 39 Glencoe Geometry

46 Parallel and Perpendicular Lines Key Points Scan the pages in the chapter and write at least one specific fact concerning each lesson. For example, in the lesson on parallel lines and transversals, one fact might be parallel planes are planes that do not intersect. After completing the chapter, you can use this table to review for your chapter test. Lesson Fact 3-1 Parallel Lines and Transversals See students work. 3-2 Angles and Parallel Lines 3-3 Adding and Subtracting Polynomials 3-4 Equations of Lines 3-5 Proving Lines Parallel 3-6 Perpendiculars and Distance Chapter 3 40 Glencoe Geometry

47 3-1 Parallel Lines and Transversals What You ll Learn Skim the Examples for Lesson 3-1. Predict two things you think you will learn about parallel lines and transversals. Sample answer: how to identify and classify 1. parallels and transversals Active Vocabulary Sample answer: how to classify angle pairs 2. New Vocabulary Use the figure to complete each statement. t Lesson 3-1 parallel lines transversal interior angles t is a transversal. 1 and 7 are exterior angles. 2 and 6 are corresponding angles a b exterior angles 4 and 5 are alternate interior angles. consecutive interior angles alternative interior angles alternate exterior angles corresponding angles a and b are parallel lines. 4 and 6 are consecutive interior angles. 2 and 7 are alternate exterior angles. 5 and 6 are interior angles. Vocabulary Link Skew is a word used in everyday English as well as in mathematics. Look up skew in the dictionary. Explain how the English definition can help you remember how skew is used in mathematics. Sample answer: turn aside or swerve; Skew lines look like they turn aside from one another. Chapter 3 41 Glencoe Geometry

48 Lesson 3-1 (continued) Main Idea Details Relationships Between Lines and Planes pp Summarize information about parallel lines and skew lines in the graphic organizer below. Parallel Lines Skew Lines What are parallel lines? coplanar lines that do not intersect What are skew lines? noncoplanar lines that do not intersect How are parallel lines and skew lines similar? Both types of lines never intersect. How are parallel lines and skew lines different? Parallel lines are on the same plane and there is a symbol,. Skew lines are on different planes and there is no symbol. Transversal Angle Pair Relationships pp Helping You Remember Model a transversal t which intersects two or more parallel lines. Identify consecutive interior angles, alternative interior and exterior angles, and corresponding angles. consecutive interior angles: t 3 and 5, 4 and 6 alternate interior angles: and 6, 4 and 5 a 3 4 alternate exterior angles: 1 and 8, 2 and corresponding angles: b and 5, 2 and 6, 3 and 7, 4 and 8 Look up meaning of the prefix trans- in the dictionary. Write down 4 words that have trans- as a prefix. How can the meaning of the prefix help you remember the meaning of transversal? Sample answer: Trans means across; transatlantic, Transamerica, transport, transparent; A transversal cuts across two or more parallel lines. Chapter 3 42 Glencoe Geometry

49 3-2 Angles and Parallel Lines What You ll Learn Scan Lesson 3-2. List two headings you would use to make an outline of this heading. Sample answer: Parallel Lines and Angle Pairs 1. Sample answer: Algebra and Angle Measures 2. Active Vocabulary parallel lines Review Vocabulary Match the term with the correct definition by drawing a line between the two. (Lesson 3-1) lie in the regions not between parallel lines that are cut by a transversal Lesson 3-2 skew interior angles exterior angles consecutive interior angles corresponding angles a line that intersects 2 or more coplanar lines at 2 different points lie on the same side of both the parallel lines and the transversal coplanar lines that never intersect lie between parallel lines that are cut by a transversal interior lines that lie on the same side of a transversal tranversal lines that are not on the same plane and never intersect Chapter 3 43 Glencoe Geometry

50 Lesson 3-2 (continued) Main Idea Details Parallel Lines and Angle Pairs pp Fill in the organizer about the Corresponding Angles Postulate. Corresponding Angles Postulate Definition If two parallel lines are cut by a transversal, each pair of corresponding angles is congruent. Why is it useful? If you know an angle measure, you will know that the corresponding angle has the same measure. Consecutive interior angles will be supplementary. Example Sample answer: t Nonexample Sample answer: t Algebra and Angle Measures pp. 180 Helping You Remember because they are corresponding angles. p q 3 and 7 are not congruent because p and q are not parallel. Fill in each blank to summarize the Perpendicular Transversal Theorem. The Perpendicular Transversal Theorem states that in a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other How can you use an everyday meaning of the adjective alternate to help you remember the types of angle pairs for two lines and a transversal? 2 4 p q Sample answer: One definition of alternate says first on one side and then the other. Alternating angles occur on opposite sides of a transversal. Chapter 3 44 Glencoe Geometry

51 3-3 Slopes of Lines What You ll Learn Scan the text in Lesson 3-3. Write two facts you learned about slopes of lines as you scanned the text. Sample answer: Slopes can be positive, negative, 1. zero, or undefined. Sample answer: Parallel lines have equal slopes. 2. Active Vocabulary Review Vocabulary Fill in each blank with the correct term or phrase. (Lesson 1-1) line made up of points with no thickness or width point a location without shape or size defined term can be explained using undefined terms undefined term can only be explained using examples and descriptions slope rate of change New Vocabulary Write the definition next to each term. ratio of the change along the y-axis to the change along the x-axis between any two points on the line same as slope; describes how a quantity y changes in relationship to a quantity x Lesson 3-3 Chapter 3 45 Glencoe Geometry

52 Lesson 3-3 (continued) Main Idea Details Slope of a Line pp Complete the organizer to summarize the steps of finding the slope of a line. Then fill in each blank of the example. Find the Slope of a Line Parallel and Perpendicular Lines pp Helping You Remember Step 1: Label the cordinates as (x 1, y 1 ) and (x 2, y 2 ). Step 2: Substitute the coordinates into the slope formula. m = y - y 2 1 x 2 - x 1 Step 3: Simplify. Find the slope of a line. Fill in each blank for the Parallel and Perpendicular Lines Postulates. Slopes of Parallel Lines: Two non-vertical lines will have the same slope if they are parallel. Vertical lines are always parallel. Slopes of Perpendicular Lines: Two nonvertical lines are perpendicular if the product of their slopes is -1. Vertical and horizontal lines are always perpendicular y x Step 1: Substitute (-2, -1) for (x 1, y 1 ) and (2, 5) for (x 2, y 2 ). Step 2: Use the formula. Step 3: Simplify. 5 - (-1) 2 - (-2) 6 4 = 3 2 Suppose 2 non-vertical lines have slopes which are reciprocals of each other. Are they perpendicular? Explain. Sample answer: Reciprocals have a product of 1, not -1 so the lines are not perpendicular. Chapter 3 46 Glencoe Geometry