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Chapter 3

Chapter 3. By Glen Harlan & Marissa Hirakawa. Lesson 1: Number Operations and Equality. The Reflexive Property a = a The Substitution Property If a = b, then a can be substituted for b in any expression. The Addition Property If a = b, then a+c = b+c The Subtraction Property

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Chapter 3

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  1. Chapter 3 By Glen Harlan & Marissa Hirakawa

  2. Lesson 1: Number Operations and Equality The Reflexive Property • a = a The Substitution Property • If a = b, then a can be substituted for b in any expression. The Addition Property • If a = b, then a+c = b+c The Subtraction Property • If a = b, then a-c = b-c The Multiplication Property • If a = b, then ac = bc The Division Property • If a = b and c  0, then a/c = b/c.

  3. Lesson 2: The Ruler and Distance Key Terms: • Coordinate: The real number which corresponds to a specific point on a line. • Distance: The real number between a pair of points on a line. • Betweenness of Points: A point is between two other points on the same line iff its coordinate is between their coordinates. The Ruler Postulate: The points on a line can be numbered so that positive number differences measure distances. The Betweenness of Points Theorem: If A-B-C, then AB + BC = AC

  4. AB + BC = AC The points on a line can be numbered so that positive number differences measure distances.

  5. Lesson 3: The Protractor and Angle Measure Key Terms: • Betweenness of Rays: A ray is between two others in the same half-rotation iff its coordinate is between their coordinates. An angle is: • acute iff it is less than 90°. • right iff it is 90°. • obtuse iff it is more than 90º but less than 180°. • straight iff it is 180°.

  6. The Protractor Postulate: The rays in a half-rotation can be numbered from 0 to 180 so that positive number differences measure angles. The Betweenness of Rays Theorem: If CA-CD-CB, then <ACD + <DCB = <ACB

  7. Lesson 4: Bisection Key Terms: • Midpoint: A point is the midpoint of a line segment iff it divides the line segment into two equal segments. • Angle Bisector: A line bisects an angle iff it divides the angle into two equal angles.

  8. Corollary to the Ruler Postulate: A line segment has exactly one midpoint. Corollary to the Protractor Postulate: An angle had exactly one ray that bisects it. Proofs Lab: • The lab in this chapter included several proof which had to be solved. • For Example: Given: <1 and <2 are complementary and m<1 = m<2 Prove: <1 and <2 are right angles

  9. Lesson 5 Complementary and Supplementary Angles

  10. 3:5 Vocab • Complementary Angles: Two angles are complementary iff their sum is 90 degrees. • In a set of complementary angles, each angle complements the other(s).

  11. Continued… • Two angles are supplementaryiff their sum is 180 degrees. • In a set of supplementary angles, each angle supplementsthe others(s).

  12. Practice with Complementary Angles

  13. Practice with Supplementary Angles

  14. Theorems • Two angles complementary to the same angle are equal to each other. • Two angles supplementary to the same angle are equal to each other.

  15. Linear Pairs and Vertical Angles Lesson 6

  16. 3:6 Vocab: • Two angles are a linear pair iff they have a common side and their other sides are opposite rays. • Two angles are vertical angles iff the sides of one angle are opposite rays to the sides of the other.

  17. Linear Pairs:

  18. Vertical Angles:

  19. Theorems • The angles in a linear pair are supplementary • Vertical angles are equal

  20. Perpendicular and Parallel Lines Lesson 7

  21. 3:7 Vocab: • Two lines are perpendiculariff they form a right angle. • Two lines are paralleliff they lie in the same plane and do not intersect.

  22. 3:7 Theorems: • Perpendicular lines form four right angles. • If the angles in a linear pair are equal, then their sides are perpendicular.

  23. PROOFS • Proofs were introduced in this chapter, in the form of fill-ins. • All definitions, theorems, and postulates from the first 3 chapters were available for chapter 3 fill-in proofs.

  24. Continued… • Algebraic Properties of Equality were also common reasons for these proofs, including the reflexive, transitive, substitution, addition, subtraction, multiplication, and division properties of equality.

  25. Example Proof O 1 2 3 D E Given: Angle 2 and Angle 3 are a linear pair Angle 1 and Angle 3 are supplementary Prove: Angle 1 = Angle 2 1: Angles 2 and 3 are a linear pair – Given 2: Angles 2 and 3 are supplementary – If 2 angles are in a linear pair, then they are supplementary 3: Angles 1 and 3 are supplementary – Given 4: Angle 1 = Angle 2 – Two angles that are supplementary to the same angle are equal.

  26. Insight (3:5-7 and Proofs) • Don’t forget that if the angles of a linear pair are equal, their sides are perpendicular. That theorem tends to be forgotten by students. • Also, don’t forget to specify the steps alluded to when using the Substitution Property of Equality. • Example: • Substitution POE (2 & 3)

  27. Continued… • Lastly, certain proofs require a “Simplify” step. This occurs when something like simple addition is used but not any specific theorem, definition, postulate, or property. • Example: • Angle TUA + Angle TAN = 90 + 90 – Addition • Angle TUA + Angle TAN = 180 - Simplify

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