1. Properties of Equality and Congruence
Section 2.6

2. Objective
Use properties of equality and congruence.

3. Key Vocabulary Reflexive Property Symmetric Property
Transitive Property
Logical Reasoning

4. Properties of Equality and Congruence
Just as in algebra where you have the properties of equality that apply to numbers, in geometry we have similar properties called the properties of congruence that apply to geometric figures.
Reflexive Property
Symmetric Property
Transitive Property
Remember
Equal (=) is used with numbers
Congruent (≅) is used with geometric figures

5. Properties of Equality and Congruence Reflexive Property

6. Reflexive Property
Jean is the same height as Jean.

7. Properties of Equality and Congruence Symmetric Property

8. Symmetric Property Jean is the same height as Pedro
Pedro is the same height as Jean If
Then

9. Properties of Equality and Congruence Transitive Property

10. Transitive Property Jean is the same height as Chris If
Jean is the same height as Pedro
And
Pedro is the same height as Chris
Then

11. Example 1 Name the property that the statement illustrates. a.
If GH  JK, then JK  GH.
DE = DE b.
If P  Q and Q  R, then P  R. c.
SOLUTION
Symmetric Property of Congruence a.
Reflexive Property of Equality b.
Transitive Property of Congruence c. 11

12. Your Turn: Name the property that the statement illustrates.
If DF = FG and FG = GH, then DF = GH. 1.
ANSWER
Transitive Property of Equality
P  P 2.
ANSWER
Reflexive Property of Congruence
If mS  mT, then mT  mS. 3.
ANSWER
Symmetric Property of Equality

13. Logical Reasoning
In geometry, you are often asked to explain why statements are true.
Logical reasoning is the system that we to explain why something is true.
Logical reasoning is the process of constructing a valid argument from observation and known facts.

14. Example 2 Definition of midpoint Definition of midpoint
In the diagram, N is the midpoint of MP, and P is the midpoint of NQ. Show that MN = PQ.
SOLUTION
MN = NP
Definition of midpoint
NP = PQ
Definition of midpoint
MN = PQ
Transitive Property of Equality 14

15. Your Turn:
1 and 2 are vertical angles, and 2  3. Show that 1  3.
1  2
2  3
1  3
Theorem
_____ ?
Given
Property of Congruence
ANSWER
Vertical Angles; Transitive

16. ALGEBRAIC PROPERTIES OF EQUALITY
Addition Property of Equality If a = b, then a + c = b + c.
Adding the same number to each side of an equation produces an equivalent equation.
Subtraction Property of Equality If a = b, then a – c = b – c.
Subtracting the same number to each side of an equation produces an equivalent equation.
Multiplication Property of Equality If a = b, then ac = bc.
Multiplying each side of an equation by the same nonzero number produces an equivalent equation.
Division Property of Equality If a = b, then a/c = b/c.
Dividing each side of an equation by the same nonzero number produces an equivalent equation.
Substitution Property of Equality If a = b, then you may Substituting a number for a variable replace b with a in any in an equation produces an expression equivalent equation.
WE USE THESE PROPERTIES TO JUSTIFY ALGEBRAIC STEPS AND SOLVE PROBLEMS. THIS IS LOGICAL REASONING

17. Example 3 1 and 2 are both supplementary to 3. Show that 1  2.
SOLUTION
Definition of supplementary angles
m1 + m3 = 180°
Definition of supplementary angles
m2 + m3 = 180°
Substitution Property of Equality
m1 + m3 = m2 + m3
Subtraction Property of Equality
m1 = m2
Definition of congruent angles
1  2 17

18. Your Turn:
Postulate
_____ ?
Distributive property
Property of Equality
Definition of
MB = AM
AB = AM + MB
AB = AM + AM
AB = 2 · AM
In the diagram, M is the midpoint of AB. Show that AB = 2 · AM.
ANSWER
midpoint; Segment Addition; Substitution

19. Joke Time What did the fish say when he hit the wall? DAM!!
What do you call 500 lawyers at the bottom of the sea.
A good start.
You're stranded in a deserted island with Attila the Hun, Adolf Hitler, and a lawyer. You have a revolver with two bullets. What do you do?
Shoot the lawyer twice!

20. Assignment
Section 2.6, pg : #1-21 odd, 25, odd