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. We can justify each step of the problem solving using postulates and properties.

Properties of equality If a = b then a + c = b + c Addition 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

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

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

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

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

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

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

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