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2.4 Reasoning with Properties from Algebra (for geometry proof)

2.4 Reasoning with Properties from Algebra (for geometry proof). ?. p. 89. What are we doing, & Why are we doing this?. In algebra, you did things because you were told to…. In geometry, we can only do what we can PROVE… We will start by justifying algebra steps (because we already know how)

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2.4 Reasoning with Properties from Algebra (for geometry proof)

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  1. 2.4 Reasoning with Properties from Algebra (for geometry proof) ? p. 89

  2. What are we doing, &Why are we doing this? • In algebra, you did things because you were told to…. • In geometry, we can only do what we can PROVE… • We will start by justifying algebra steps (because we already know how) • Then we will continue justifying steps into geometry…

  3. But first…we need to 1. Learn the different properties / justifications 2. Know format for proving / justifying mathematical statements 3. Apply geometry properties to proofs Here’s an example of a proof…

  4. Properties of Equality(from algebra) • Addition property of equality- if a=b, then a+c=b+c. (can add the same #, c, to both sides of an equation) • Subtraction property of equality - If a=b, then a-c=b-c. (can subtract the same #, c, from both sides of an equation) • Multiplication prop. of equality- if a=b, then ac=bc. • Division prop. of equality- if a=b, then

  5. Properties of equality (algebra) • Reflexive prop. of equality- a=a • Symmetric prop of equality- if a=b, then b=a. • Transitive prop of equality- if a=b and b=c, then a=c. • Substitution prop of equality- if a=b, then a can be plugged in for b and vice versa. • Distributive prop.- a(b+c)=ab+ac OR (b+c)a=ba+ca

  6. Properties of equality (geometry) (mirror) • Reflexive Property AB ≅ AB ∠ A ≅∠ A • Symmetric Property If AB ≅ CD, then CD ≅ AB If ∠ A ≅∠ B, then ∠ B ≅∠ A • Transitive Property If AB ≅ CD and CD ≅ EF, then AB ≅ EF If ∠ A ≅∠ B and ∠ B ≅∠ C,then ∠ A ≅∠ C (twins) (triplets)

  7. 2(3x+1) = 5x+14 6x+2 = 5x+14 x+2 = 14 x = 12 Given Distributive prop Subtraction prop of = Subtraction prop of = Ex: Solve the equation & write a reason for each step.

  8. 55z-3(9z+12) = -64 55z-27z-36 = -64 28z-36 = -64 28z = -28 z = -1 Given Distributive prop Simplify (or collect like terms) Addition prop of = Division prop of = Solve 55z-3(9z+12) = -64 & write a reason for each step.

  9. Solving an Equation in Geometry with justifications NO = NM + MO Segment Addition Post. 4x – 4 = 2x + (3x – 9) Substitution Property of Equality 4x – 4 = 5x – 9 Simplify. –4 = x – 9 Subtraction Property of Equality 5 = x Addition Property of Equality

  10. Solve, Write a justification for each step.  Add. Post. mABC = mABD + mDBC 8x°=(3x + 5)°+(6x – 16)° Subst. Prop. of Equality 8x = 9x – 11 Simplify. –x = –11 Subtr. Prop. of Equality. x = 11 Mult. Prop. of Equality.

  11. Remember! Numbers are equal (=) and figures are congruent ().

  12. Identifying Property of Equality and Congruence Identify the property that justifies each statement. A. QRS  QRS B. m1 = m2 so m2 = m1 C. AB  CD and CD  EF, so AB EF. D. 32° = 32° Reflex. Prop. of . Symm. Prop. of = Trans. Prop of  Reflex. Prop. of =

  13. Other “Justifications” Defn of…

  14. Example from scratch…

  15. Classwork - Go Page 91 #’s 1-23 odd, 27-30 Read Pg 95

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