2. Page 53, Chapter Summary: Concepts and Procedures After studying this CHAPTER, you should be able to... 1.1 Recognize points, lines, segments, rays, angles, and triangles. 1.2 Measure segments and angles 1.2 Classify angles and name the parts of a degree 1.2 Recognize congruent angles and segments 1.3 Recognize collinear and noncollinear points 1.3 Recognize when a point is between two other points 1.3 Apply the triangle inequality principle 1.3 Correctly interpret geometric diagrams 1.4 Write simple two-column proofs 1.5 Identify bisectors and trisectors of segments and angles 1.6 Write paragraph proofs 1.7 Recognize that geometry is based on a deductive structure 1.7 Identify undefined terms, postulates, and definitions 1.7 Understand the characteristics and application of theorems 1.8 Recognize conditional statements and the negation, the converse, the inverse, and the contrapositive of a statement 1.8 Use the chain-rule to draw conclusions 1.9 Solve probability problems 2

3. Chapter 1, Section 1: “Getting Started” 1.1 Recognize points, lines, segments, rays, angles, and triangles. ANGLE INTERSECTION LINELINE SEGMENT NUMBER LINE POINT RAYSEGMENT TRIANGLE UNION VERTEX After studying this SECTION, you should be able to... Related Vocabulary ENDPOINTS 3

4. 1.1 Recognize points, lines, segments, rays, angles, and triangles. ANGLE LINELINE SEGMENT POINT SEGMENT After studying this SECTION, you should be able to... Related Vocabulary ENDPOINTS Chapter 1, Section 1: “Getting Started” 4

5. 1.1 Recognize points, lines, segments, rays, angles, and triangles. INTERSECTION NUMBER LINE RAYTRIANGLE UNION VERTEX After studying this SECTION, you should be able to... Related Vocabulary 0123- 3- 2- 1 Chapter 1, Section 1: “Getting Started” 5

6. Your Turn! What’s My Name? 1. Q 2. 3. 4. 6 A T C G O D EB point Qor Q ray CA or ray CT CACT line DG …or line DO, GD, GO, or OD GOGDDODG segment BE or segment EB EB BE To see answers, hit space bar. OD Easy peasy!

7. After studying this SECTION, you should be able to... 1.2 Measure segments and angles 1.2 Classify angles and name the parts of a degree 1.2 Recognize congruent angles and segments ACUTE ANGLE CONGRUENT ANGLES CONGRUENT SEGMENTS MEASURE MINUTES OBTUSE ANGLE PROTRACTOR RIGHT ANGLE SECONDSDEGREES STRAIGHT ANGLE TICK MARK Related Vocabulary Chapter 1, Section 2: “Measurement of Segments and Angles” 7

8. After studying this SECTION, you should be able to... 1.2 Measure segments and angles 1.2 Classify angles and name the parts of a degree 1.2 Recognize congruent angles and segments ACUTE ANGLEOBTUSE ANGLERIGHT ANGLESTRAIGHT ANGLE Related Vocabulary Chapter 1, Section 2: “Measurement of Segments and Angles” 8

9. After studying this SECTION, you should be able to... 1.2 Measure segments and angles 1.2 Classify angles and name the parts of a degree 1.2 Recognize congruent angles and segments CONGRUENT ANGLESCONGRUENT SEGMENTS Related Vocabulary Chapter 1, Section 2: “Measurement of Segments and Angles” 9

10. After studying this SECTION, you should be able to... 1.2 Measure segments and angles 1.2 Classify angles and name the parts of a degree 1.2 Recognize congruent angles and segments MEASURE Degrees & MINUTES PROTRACTORRULER Degrees, Minutes, & SECONDS DEGREES TICK MARK Related Vocabulary Chapter 1, Section 2: “Measurement of Segments and Angles” 360 ⁰ 359 ⁰ 60’359 ⁰ 59’ 60” 10

11. 1.3 Recognize collinear and noncollinear points 1.3 Recognize when a point is between two other points 1.3 Apply the triangle inequality principle 1.3 Correctly interpret geometric diagrams COLLINEARNONCOLLINEAR Related Vocabulary After studying this SECTION, you should be able to... Chapter 1, Section 3: “Collinearity, Betweenness, and Assumptions” XYZ X Z Y X, Y, and Z are collinear X, Y, and Z are NOT collinear 11

12. 1.3 Recognize collinear and noncollinear points 1.3 Recognize when a point is between two other points 1.3 Apply the triangle inequality principle 1.3 Correctly interpret geometric diagrams COLLINEARITY  Betweenness of Points Related Vocabulary After studying this SECTION, you should be able to... Chapter 1, Section 3: “Collinearity, Betweenness, and Assumptions” XYZ Y is between X and Z X Z Y Y is NOT between X and Z 12

13. 1.3 Recognize collinear and noncollinear points 1.3 Recognize when a point is between two other points 1.3 Apply the triangle inequality principle 1.3 Correctly interpret geometric diagrams POSTULATE: After studying this SECTION, you should be able to... Chapter 1, Section 3: “Collinearity, Betweenness, and Assumptions” The sum of the measures of any two sides of a triangle is always greater than the measure of the third side. Nope! YES! 13

14. 1.3 Recognize collinear and noncollinear points 1.3 Recognize when a point is between two other points 1.3 Apply the triangle inequality principle 1.3 Correctly interpret geometric diagrams TRIANGLE INEQUALITY: After studying this SECTION, you should be able to... Chapter 1, Section 3: “Collinearity, Betweenness, and Assumptions” For any three points, there are only two possibilities: YES! 1.They are collinear (one point is between the other two, such that two of the measures equals the third, or 2. They are noncollinear (the three points determine a triangle! The sum of any two sides exceeds the measure of the third side! 14

15. 1.3 Recognize collinear and noncollinear points 1.3 Recognize when a point is between two other points 1.3 Apply the triangle inequality principle 1.3 Correctly interpret geometric diagrams See very important TABLE on page 19! After studying this SECTION, you should be able to... Chapter 1, Section 3: “Collinearity, Betweenness, and Assumptions” Do Assume: AD and BE are straight lines ∡ BCE is a straight angle C, D, and E are noncollinear C is between B and E E is to the right of A A C D E B Allowable Assumptions: Straight lines Straight angles Noncollinearity Betweenness of points Relative position of points 15

16. 1.3 Recognize collinear and noncollinear points 1.3 Recognize when a point is between two other points 1.3 Apply the triangle inequality principle 1.3 Correctly interpret geometric diagrams See very important TABLE on page 19! After studying this SECTION, you should be able to... Chapter 1, Section 3: “Collinearity, Betweenness, and Assumptions” DO NOT Assume: AB ≅ CD ∡ BAC is a right angle ∡B ≅ ∡E ∡ CDE is an obtuse angle BC is longer than CE A C D E B Forbidden Assumptions: Right Angles Congruent segments Congruent Angles Relative SIZES of segments Relative SIZES of angles 16 You must PROVE these!

17. 1.4 Write simple two-column proofs After studying this SECTION, you should be able to... Related Vocabulary THEOREM TWO-COLUMN PROOF Chapter 1, Section 4: “Beginning Proofs” - a mathematical statement that can be proved #4 Statements#5 Reasons - A step-by-step logical argument offering proof by a chain of statements and reasons in support of a specific conclusion. A two-column proof has FIVE parts:1. Givens 2. Prove 3. Diagram Example, Theorem 1: If two angles are right angles, then they are congruent. #1 Given: ∡A is a right ∡ ∡B is a right ∡ #2 Prove ∡A ≅ ∡B #3 Diagram A B 1. ∡A is a right ∡ 3. ∡B is a right ∡ 2. m ∡A = 90 4. m ∡B = 90 5. ∡A ≅ ∡B 1. Given 2. If an ∡ is a right ∡, then its measure is 90 3. Given 4. Same as #2 5. If 2 ∡ ’s have the same measure, then they are ≅ 17

18. 1.5 Identify bisectors and trisectors of segments and angles Related Vocabulary After studying this SECTION, you should be able to... BISECTBISECTOR MIDPOINT TRISECTTRISECTORS TRISECTION POINTS Chapter 1, Section 5: “Division of Segments and Angles” 18

19. 1.5 Identify bisectors and trisectors of segments and angles Related Vocabulary After studying this SECTION, you should be able to... BISECT BISECTORMIDPOINT Chapter 1, Section 5: “Division of Segments and Angles” (verb) to divide into two congruent parts (noun) the POINT that divides a segment into two congruent segments (noun) the name of the point that divides a segment into two congruent segments Question: Is it possible for a line to have a MIDPOINT? Question: How would you know if the segment above had been TRISECTED ? 19

20. 1.5 Identify bisectors and trisectors of segments and angles Related Vocabulary After studying this SECTION, you should be able to... BISECT BISECTOR Chapter 1, Section 5: “Division of Segments and Angles” (verb) to divide into two congruent parts (noun) the RAY that divides an angle into two congruent angles Question: Is it possible for an angle to have a MIDPOINT? Question: How would you know the angle above had been TRISECTED ? 20

21. 1.5 Identify bisectors and trisectors of segments and angles Related Vocabulary After studying this SECTION, you should be able to... TRISECT - TRISECTORS TRISECTION POINTS Chapter 1, Section 5: “Division of Segments and Angles” (verb) to divide a segment or angle into THREE congruent parts. 21

22. 1.6 Write paragraph proofs Related Vocabulary After studying this SECTION, you should be able to... COUNTEREXAMPLE - PARAGRAPH PROOF - Chapter 1, Section 6: “Paragraph Proofs” Facts that are inconsistent with theory – or an argument proving that a fact, hypothesis or mathematical theorem is not true. NOTE: This is an introduction to the paragraph method of proof. We will use the Paragraph form exclusively when we get to Indirect Proofs in Chapter 5. While paragraph proofs can also be used to prove a mathematical conclusion, you will mostly rely upon the two- column method to do so in this course. When writing an “Indirect Proof” in paragraph form, you will be attempting to arrive at a conclusion by proving the alternative to it false. Therefore, “Indirect Proof” can also be referred to as “Proof by Contradiction.” Has THREE parts: * Introduction * Body * Conclusion Like any good paper, 22

23. 1.7 Recognize that geometry is based on a deductive structure 1.7 Identify undefined terms, postulates, and definitions 1.7 Understand the characteristics and application of theorems Related Vocabulary After studying this SECTION, you should be able to... CONCLUSION CONDITIONAL STATEMENT CONVERSE DEDUCTIVE STRUCTURE DEFINITION HYPOTHESIS IMPLICATION POSTULATE Chapter 1, Section 7: “Deductive Structure” 23

24. 1.7 Recognize that geometry is based on a deductive structure 1.7 Identify undefined terms, postulates, and definitions 1.7 Understand the characteristics and application of theorems Related Vocabulary After studying this SECTION, you should be able to... DEDUCTIVE STRUCTURE DEFINITION POSTULATE Chapter 1, Section 7: “Deductive Structure” – conclusions are supported and proved by using allowable assumptions and statements that have been proved to be true. Deductive reasoning – the process of drawing a conclusion based on logical or reasonable information or facts. Inductive reasoning – reaching a conclusion based on observation alone. Generalizing. – an unproved assumption. – states the meaning of a term or idea. UNDEFINED TERMS – terms that are described. Example: points and lines 24 Use these + theorems in proofs!

25. 1.7 Recognize that geometry is based on a deductive structure 1.7 Identify undefined terms, postulates, and definitions 1.7 Understand the characteristics and application of theorems Related Vocabulary After studying this SECTION, you should be able to... CONCLUSION - CONDITIONAL STATEMENT - HYPOTHESIS - IMPLICATION Chapter 1, Section 7: “Deductive Structure” DECLARATIVE STATEMENT - (definition) – a midpoint is a point that divides a segment (or an arc) into two congruent parts If a point is the midpoint of a segment, then the point divides the segment into two congruent segments  CONDITIONAL STATEMENT “If a point is the midpoint of a segment, The “ If...,” clause of the conditional statement  The “ then...” clause of the conditional statement  then the point divides the segment into two congruent segments.”  “If..., then...” 25

26. 1.7 Recognize that geometry is based on a deductive structure 1.7 Identify undefined terms, postulates, and definitions 1.7 Understand the characteristics and application of theorems Related Vocabulary After studying this SECTION, you should be able to... CONCLUSION - CONDITIONAL STATEMENT CONVERSE - HYPOTHESIS - IMPLICATION Chapter 1, Section 7: “Deductive Structure” If p, then q If p, then q If q, then p Let p = “If a point is the midpoint of a segment,” Let q = “then the point divides the segment into two congruent segments” If a point divides a segment into two congruent segments, then the point is the midpoint of the segment Reversing the hypothesis and conclusion In this definition, the hypothesis and conclusion are reversible. If a definition is a GOOD definition, it is always reversible! 26

27. 1.7 Recognize that geometry is based on a deductive structure 1.7 Identify undefined terms, postulates, and definitions 1.7 Understand the characteristics and application of theorems Related Vocabulary After studying this SECTION, you should be able to... CONCLUSION - CONDITIONAL STATEMENT CONVERSE - HYPOTHESIS - IMPLICATION Chapter 1, Section 7: “Deductive Structure” Theorem 1: If two angles are right angles, then they are congruent If p, then q If q, then p Let p = “If two angles are right angles,” Let q = “then they are congruent” If two angles are congruent, then they are right angles. The converse is FALSE! Postulates and theorems are NOT always reversible, unlike GOOD definitions! Reversing the hypothesis and conclusion 27

28. If you write a definition and find it is false when reversed, then what you wrote is NOT a GOOD definition! 28

29. 1.8 Recognize conditional statements 1.8 Use the chain-rule to draw conclusions Related Vocabulary After studying this SECTION, you should be able to... CONTRAPOSITIVE INVERSE CHAIN RULE NEGATION VENN DIAGRAM Chapter 1, Section 8: “Statements of Logic” Also, from 1.7 Declarative sentence Conditional sentence Hypothesis Conclusion Implication 1.8 Recognize the negation of a statement 1.8 Recognize the converse, the inverse, and the contrapositive of a statement 29

30. 1.8 Recognize conditional statements 1.8 Use the chain-rule to draw conclusions Related Vocabulary After studying this SECTION, you should be able to... NEGATION - Chapter 1, Section 8: “Statements of Logic” Declarative sentence Conditional sentence Hypothesis Conclusion Implication 1.8 Recognize the negation of a statement 1.8 Recognize the converse, the inverse, and the contrapositive of a statement Two straight angles are congruent If two angles are straight angles, then they are congruent If two angles are straight angles, then they are congruent If p, then q Symbols: p  q Words: p implies q Words: “not p” Symbols: ~ p To contradict or state the opposite of something 30

31. 1.8 Recognize conditional statements 1.8 Use the chain-rule to draw conclusions Related Vocabulary After studying this SECTION, you should be able to... CONTRAPOSITIVE INVERSE CONVERSE VENN DIAGRAM Chapter 1, Section 8: “Statements of Logic” 1.8 Recognize the negation of a statement 1.8 Recognize the converse, the inverse, and the contrapositive of a statement Conditional “if p, then q”: If you live in Lexington, then you live in Kentucky. If q, then p: If you live in Kentucky, then you live in Lexington. If ~p, then ~q If you don’t live in Lexington, then you don’t live in Kentucky. If ~q, then ~p If you don’t live in Kentucky, then you don’t live in Lexington. Kentucky Lexington To determine the truth value of each statement, we must first assume that the original conditional statement is TRUE. FALSE! F A L S E ! TRUE! If the conditional statement is TRUE, then the contrapositive will always be TRUE! 31

32. 1.8 Recognize conditional statements 1.8 Use the chain-rule to draw conclusions Related Vocabulary After studying this SECTION, you should be able to... CHAIN RULE Chapter 1, Section 8: “Statements of Logic” 1.8 Recognize the negation of a statement 1.8 Recognize the converse, the inverse, and the contrapositive of a statement - The logical sequence you follow when writing proofs Words: If p implies q, and q implies r, then p implies r. Symbols: If p  q, and q  r, then p  r. Mathematically: since 5 = 5,... then x = y In a Proof: If ∡ X is a right angle and ∡ Y is a right angle, then ∡ X ≅ ∡ Y and all right angles equal 90, 32

33. 1.9 Solve probability problems Related Vocabulary After studying this SECTION, you should be able to... PROBABILITY - Chapter 1, Section 9: “Probability” The chance that something will happen Favorable PART TOTAL Possibilities A ratio whose value is between 0 and 1, inclusive. : ImpossibleCertain 01½ Equally Likely Less likely More Likely STEPS: 1. List ALL outcomes 2.Record “winners” over total 33

34. C B A D If three of the four points are selected in a random order, what is the probability that the ordered letters will correctly name the angle shown? LIST all possibilities: ABC ABD ACB ACD ADB ADC BAC BAD BCA BCD BDA BDC CAB CAD CBA CBD CDA CDB DAB DAC DBA DBC DCA DCB Or use the Fundamental Counting Principle: 432 # of ways to select the first point # of ways to select the second point # of ways to select the third point TOTAL 24 34

35. C B A D If three of the four points are selected in a random order, what is the probability that the ordered letters will correctly name the angle shown? Circle the “winners”: ABC ABD ACB ACD ADB ADC BAC BAD BCA BCD BDA BDC CAB CAD CBA CBD CDA CDB DAB DAC DBA DBC DCA DCB PART 4 Answer: Part TOTAL 24 4 6 1 Don’t forget to REDUCE! 35