1. 2.5 and 2.6 Properties of Equality and Congruence

2. Objective: You will use deductive reasoning to:
Write proofs using geometric theorem
To use algebraic properties in logical arguments.

3. Algebraic Properties
Substitution Property: If a = b, then a can be substituted for b in an equation or expression.
Distributive Property: a(b + c) = ab + ac, where a, b, and c are real numbers.

4. Algebraic Properties Addition Property: If a = b, then a + c = b + c.
Subtraction Property:
If a = b, then a – c = b – c.
Multiplication Property:
If a = b, then ac = bc.
Division Property:
If a = b and c = 0, then a/c = b/c.

5. Example 1: Write a two-column proof to solve the equation.
Statements Reasons
3x + 2 = 8
3x + 2 – 2 = 8 – 2
3x = 6
3x ÷ 3 = 6 ÷ 3
x = 2
1. Given
2. Subtraction Prop
3. Simplify
4. Division Prop
5. Simplify

6. Example 2: Write a two-column proof to solve the equation.
Statements Reasons
4x + 9 = 16 – 3x
4x x = 16 – 3x + 3x
7x + 9 = 16
7x + 9 – 9 = 16 – 9
7x = 7
7x ÷ 7 = 7 ÷ 7
x = 1
Given
Addition Prop
Simplify
Subtraction Prop
Division Prop

8. Example 3: Write a two-column proof to solve the equation.
Statements Reasons
2(-x – 5) = 12 Given
-2x – 10 = 12 Distributive Prop
-2x – = Addition Prop
-2x = Simplify
-2x ÷ -2 = 22 ÷ Division Prop
x = Simplify

10. Reflexive Property
Equality
Congruence

12. Symmetric Property
Equality:
Congruence:

14. Transitive Property
Equality:
Congruence:

15. Properties of Equality
Addition Property: adding a number to each side of an equation
Subtraction Property: subtracting a number from each side of an equation

16. Properties of Equality
Multiplication Property: multiplying by a number on each side of an equation
Division Property: dividing by a number on each side of an equation
Substitution Property: substituting a number for a variable in an equation to produce an equivalent equation

17. Definitions
Theorem:
A true statement that follows as a result of other true statements.
Two-column proof:
Most commonly used. Has numbered statements and reasons that show the logical order of an argument.

18. Use the diagram and the given information to complete the missing steps and reasons in the proof.
GIVEN: LK = 5, JK = 5, JK ≅ JL
PROVE: LK ≅ JL
Statements: Reasons:
_______________
LK = JK
LK ≅ JK
JK ≅ JL
________________
Given
Transitive Property
_______________