Download Properties of Equality and Congruence - peacock

Properties of
Equality and
Congruence
Section 2.6
Objective
• Use properties of equality and
congruence.
Key Vocabulary
•
•
•
•
Reflexive Property
Symmetric Property
Transitive Property
Logical Reasoning
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
Properties of Equality and Congruence
Reflexive Property
Equality
AB  AB
mA  mB
Congruence
AB  AB
A  A
Reflexive Property
Jean is the same
height as Jean.
Properties of Equality and Congruence
Symmetric Property
Equality
If AB  CD, then CD  AB
If mA  mB, then mB  mA
Congruence
If AB  CD, then CD  AB
If A  B, then B  A
Symmetric Property
If Jean is the same
height as Pedro
Then
Pedro is the same
height as Jean
Properties of Equality and Congruence
Transitive Property
Equality
If AB  CD and CD  EF ,
then AB  EF .
If mA  mB and mB  mC ,
then mA  mC.
Congruence
If AB  CD and CD  EF ,
then AB  EF .
If A  B and B  C ,
then A  C.
Transitive Property
If
Jean is the same
height as Pedro
And
Pedro is the same Then
height as Chris
Jean is the same
height as Chris
Example 1
Name the property that the statement illustrates.
a. If GH  JK, then JK  GH.
b. DE = DE
c. If P  Q and Q  R, then P  R.
SOLUTION
a. Symmetric Property of Congruence
b. Reflexive Property of Equality
c. Transitive Property of Congruence
Your Turn:
Name the property that the statement illustrates.
1. If DF = FG and FG = GH, then DF = GH.
ANSWER
Transitive Property of Equality
2. P  P
ANSWER
Reflexive Property of Congruence
3. If mS  mT, then mT  mS.
ANSWER
Symmetric Property of Equality
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.
Example 2
In the diagram, N is the midpoint of MP, and P is the
midpoint of NQ. Show that MN = PQ.
SOLUTION
MN = NP
NP = PQ
MN = PQ
Definition of midpoint
Definition of midpoint
Transitive Property of Equality
Your Turn:
1 and 2 are vertical angles, and 2  3. Show
that 1  3.
1  2
?
_____
Theorem
2  3
Given
1  3
_____
?
Property of Congruence
ANSWER
Vertical Angles; Transitive
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
Example 3
1 and 2 are both supplementary to 3. Show
that 1  2.
SOLUTION
m1 + m3 = 180°
Definition of supplementary angles
m2 + m3 = 180°
Definition of supplementary angles
m1 + m3 = m2 + m3
Substitution Property of Equality
m1 = m2
Subtraction Property of Equality
1  2
Definition of congruent angles
Your Turn:
In the diagram, M is the midpoint of AB. Show that
AB = 2 · AM.
MB = AM
AB = AM + MB
AB = AM + AM
AB = 2 · AM
ANSWER
?
Definition of _____
_____
?
Postulate
_____
?
Property of Equality
Distributive property
midpoint; Segment Addition; Substitution
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!
Assignment
• Section 2.6, pg. 91-94: #1-21 odd, 25, 2935 odd