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Chapter 1 Overview: Introduction to Geometry

Chapter 1 Overview: Introduction to Geometry. 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.

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Chapter 1 Overview: Introduction to Geometry

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  1. Chapter 1 Overview: Introduction to Geometry

  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

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

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

  5. Chapter 1, Section 1: “Getting Started” After studying this SECTION, you should be able to . . . 1.1 Recognize points, lines, segments, rays, angles, and triangles. Related Vocabulary UNION RAY TRIANGLE VERTEX INTERSECTION NUMBER LINE - 3 - 2 - 1 0 1 2 3

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

  7. Chapter 1, Section 2: “Measurement of Segments and Angles” 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 Related Vocabulary ACUTE ANGLE OBTUSE ANGLE RIGHT ANGLE STRAIGHT ANGLE CONGRUENT ANGLES CONGRUENT SEGMENTS MEASURE DEGREES MINUTES SECONDS PROTRACTOR TICK MARK

  8. Chapter 1, Section 2: “Measurement of Segments and Angles” 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 Related Vocabulary ACUTE ANGLE OBTUSE ANGLE RIGHT ANGLE STRAIGHT ANGLE

  9. Chapter 1, Section 2: “Measurement of Segments and Angles” 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 Related Vocabulary CONGRUENT ANGLES CONGRUENT SEGMENTS

  10. Chapter 1, Section 2: “Measurement of Segments and Angles” 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 Related Vocabulary PROTRACTOR RULER MEASURE TICK MARK 360⁰ 359⁰ 60’ 359⁰ 59’ 60” Degrees, Minutes, & SECONDS DEGREES Degrees & MINUTES

  11. Chapter 1, Section 3: “Collinearity, Betweenness, and Assumptions” After studying this SECTION, you should be able to . . . 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 Related Vocabulary X, Y, and Z are NOT collinear COLLINEAR NONCOLLINEAR Z X X Y Z X, Y, and Z are collinear Y

  12. Chapter 1, Section 3: “Collinearity, Betweenness, and Assumptions” After studying this SECTION, you should be able to . . . 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 Related Vocabulary COLLINEARITY  Betweenness of Points Y is NOT between X and Z Z X X Y Z Y is between X and Z Y

  13. Chapter 1, Section 3: “Collinearity, Betweenness, and Assumptions” After studying this SECTION, you should be able to . . . 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: The sum of the measures of any two sides of a triangle is always greater than the measure of the third side. Nope! Nope! YES!

  14. Chapter 1, Section 3: “Collinearity, Betweenness, and Assumptions” After studying this SECTION, you should be able to . . . 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: For any three points, there are only two possibilities: • 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! YES! The sum of any two sides exceeds the measure of the third side!

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

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

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

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

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

  20. Chapter 1, Section 5: “Division of Segments and Angles” After studying this SECTION, you should be able to . . . 1.5 Identify bisectors and trisectors of segments and angles Related Vocabulary BISECTOR BISECT (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 ?

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

  22. Chapter 1, Section 6: “Paragraph Proofs” After studying this SECTION, you should be able to . . . 1.6 Write paragraph proofs Related Vocabulary COUNTEREXAMPLE - Facts that are inconsistent with theory – or an argument proving that a fact, hypothesis or mathematical theorem is not true. PARAGRAPH PROOF - 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. Like any good paper, 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. Has THREE parts: * Introduction When writing an “Indirect Proof” in paragraph form, you will be attempting to arrive at a conclusion by proving the alternative to it false. * Body * Conclusion Therefore, “Indirect Proof” can also be referred to as “Proof by Contradiction.”

  23. Chapter 1, Section 7: “Deductive Structure” After studying this SECTION, you should be able to . . . 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 CONVERSE CONDITIONAL STATEMENT DEDUCTIVE STRUCTURE IMPLICATION DEFINITION HYPOTHESIS CONCLUSION POSTULATE

  24. Chapter 1, Section 7: “Deductive Structure” After studying this SECTION, you should be able to . . . 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 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. DEDUCTIVE STRUCTURE – conclusions are supported and proved by using allowable assumptions and statements that have been proved to be true. UNDEFINED TERMS – terms that are described. Use these + theorems in proofs! Example: points and lines POSTULATE – an unproved assumption. DEFINITION – states the meaning of a term or idea.

  25. Chapter 1, Section 7: “Deductive Structure” After studying this SECTION, you should be able to . . . 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 DECLARATIVE STATEMENT - (definition) – a midpoint is a point that divides a segment (or an arc) into two congruent parts CONDITIONAL STATEMENT - If a point is the midpoint of a segment, then the point divides the segment into two congruent segments IMPLICATION  CONDITIONAL STATEMENT  “If . . ., then . . .” HYPOTHESIS - The “If. . .,” clause of the conditional statement  “If a point is the midpoint of a segment, CONCLUSION - The “then . . .” clause of the conditional statement  then the point divides the segment into two congruent segments.”

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

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

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

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

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

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

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

  33. Chapter 1, Section 9: “Probability” After studying this SECTION, you should be able to . . . 1.9 Solve probability problems Related Vocabulary PROBABILITY - The chance that something will happen STEPS: 1. List ALL outcomes A ratio whose value is between 0 and 1, inclusive. : Favorable PART • Record “winners” • over total TOTAL Possibilities 0 ½ 1 Impossible Equally Likely Certain Less likely More Likely

  34. 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? TOTAL A B 24 C D LIST all possibilities: Or use the Fundamental Counting Principle: BAC CAB DAB ABC ABD BAD CAD DAC BCA ACB CBA DBA 4 3 2 BCD ACD CBD DBC BDA # of ways to select the second point # of ways to select the third point ADB CDA DCA # of ways to select the first point BDC ADC CDB DCB

  35. 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? PART A B 4 Don’t forget to REDUCE! C D Answer: Circle the “winners”: BAC CAB DAB ABC ABD BAD 4 Part 1 CAD DAC BCA ACB CBA DBA BCD ACD CBD DBC 6 24 TOTAL BDA ADB CDA DCA BDC ADC CDB DCB

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