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Section 2.4: Reasoning in Algebra

Section 2.4: Reasoning in Algebra. Objective: To connect reasoning in algebra and geometry. Reasoning in algebra. In Geometry, we accept postulates and properties as true. We use properties of equality to solve problems.

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Section 2.4: Reasoning in Algebra

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  1. Section 2.4: Reasoning in Algebra Objective: To connect reasoning in algebra and geometry

  2. Reasoning in algebra • In Geometry, we accept postulates and properties as true. • We use properties of equality to solve problems. • We can justify each step of the problem solving using postulates and properties.

  3. Properties of equality • If a = b then a + c = b + cAddition Property of Equality • If a = b then a - c = b – c Subtraction Property of Equality • If a = b, then a ● c = b ● c Multiplication Property of Equality • If a = b, then , c ≠ 0 Division Property of Equality • a = a Reflexive Property of Equality • If a = b, then b = a Symmetric Property of Equality • If a = b and b = c, then a = c Transitive Property of Equality

  4. More properties of equality • Substitution Property: • If a = b, then b can replace a in any expression • The Distributive Property: • a(b + c) = ab + bc

  5. Acceptable justifications (Why is each step of a problem true??): • Given Statements • Postulates • Properties of Equality or Congruence • Definitions

  6. Example • Use the figure to solve for x. Justify each step. • Given: AC = 21 • 15-x 4+2x • AB + BC = AC • 15-x + (4+2x) = 21 • 19+x= 21 • x=2

  7. Example • Solve for x and justify each step. • Given m ABC = 128º • m ABD + m DBC = m ABC • x + 2x + 5 = 128 • 3x + 5 = 128 • 3x = 123 • x = 41

  8. Properties of congruence • 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

  9. Using Properties of equality and congruence • Name the property that justifies each statement. • If x = y and y + 4 = 3x, then x + 4= 3x • If x + 4 = 3x, then 4 = 2x • If

  10. Equality vs. Congruence • Congruence: • Compares 2 geometric shapes • and • then • TRANSITIVE PROPERTY OF CONGRUENCE • (Segments are same size) • Equality: • Compares 2 quantities • AB = CD and CD = EF, then • AB = EF • TRANSITIVE PROPERTY OF EQUALITY • (the lengths are equal)

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