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 + 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

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 = x 4+2x AB + BC = AC 15-x + (4+2x) = 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
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)