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Section 2.4: Properties of Equality and Algebraic Proofs. October 5, 2009. Addition. If a=b, then a + c = b + c If x – 5 = 7, then x = 12 Since x – 5 + 5 = 7 + 5 This can be used with Angles and Segments as well If <A = <B , then <A + 90 = <B + 90. Subtraction.

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addition
Addition
  • If a=b, then a + c = b + c
  • If x – 5 = 7, then x = 12
    • Since x – 5 + 5 = 7 + 5
  • This can be used with Angles and Segments as well
    • If <A = <B , then <A + 90 = <B + 90
subtraction
Subtraction
  • If a=b, then a - c = b - c
  • If x + 3 = 10, then x = 7
    • Since x + 3 - 3 = 10 - 3
  • This can be used with Angles and Segments as well
    • If <A + <B = 180, then <A = 180 - <B
      • Since <A + <B - <B = 180 - <B
multiplication
Multiplication
  • If a=b, then ac = bc
  • If ½x = 5, then x = 10
    • Since 2(½x) = 2(5)
  • This can be used with Angles and Segments as well
    • If <A = 45, then 2(<A) = 90
      • Since 2(<A) = 2(45)
division
Division
  • If a=b, then a / c = b / c (if c is not 0)
  • If 4x = 12, then x = 3
    • Since 4x / 4 = 12 / 4
  • This can be used with Angles and Segments as well
    • If 2AB = 20, then AB = 10
      • Since 2AB / 2 = 20 / 2
distributive
Distributive
  • If a ( b + c), then ab + ac
  • If 2( x + 6) = 10, then 2x + 12 = 10
  • This can be used with Angles and Segments as well
    • If 3(AB + BC) = 21, then 3AB + 3BC = 21
substitution
Substitution
  • If a = b and a = c, then b = c
  • If x + 5 = y and x = 4, then y = 9
    • Since 4 + 5 = y, so 9 = y and y = 9
  • This can be used with Angles and Segments as well
    • If <A + <B = 180 and <B = <C,

then <A + <C = 180

reflexive symmetric transitive
Reflexive, Symmetric, Transitive
  • Reflexive:
      • a = a
      • x = x
      • <A = <A
  • Symmetric:
      • If a = b, then b = a
      • If x + 5 = 15, then 15 = x + 5
      • If 90 = <A, then <A = 90
  • Transitive:
      • If a = b and b = c, then a = c
      • If x + 5 = y, and y = 10, then x + 5 = 10
      • If <A + <B = <C and <C = 90, then

<A + <B = 90

practice name the property of equality
Practice – Name the Property of Equality
  • If x + 5 = -11, then x = -16
  • If AB+BC=AC and AC=10, then AB+BC=10
  • If <C=90 and <B=90, then <C=<B
  • If 3x + 5= 12, then x + 5/3 = 4
  • If -2/3x = 4, then x = -6
proofs
Proofs
  • Formal Proof – a two column proof of statements and reason which follows a step-by-step procedure to reach a conclusion. THESE ARE THE KIND WE WILL USE.
  • There are also:
    • Informal Proofs, which are a paragraph describing each step.
    • Flow Proof, which show each step in a flow chart.
formal proofs
Formal Proofs

LOOK AT THE PROOFS ON PW 2-4.

You will always be given 2 things in a Formal Proof.

  • Given: This is your 1st statement
  • Prove: This is your last statement

Statement #2 is always based on what you are given.

Statements #3-5 are based on what additional things you need for your prove statement.