2.6 Properties of Equality and
Congruence
Goal
Use properties of equality and congruence.
REFLEXIVE PROPERTY
Equality
Congruence
AB AB
maA maA
AB
& c AB
&
aA c aA
SYMMETRIC PROPERTY
Equality
Congruence
If AB CD, then CD AB.
If AB
& c CD
&, and CD
& c AB
&.
If maA maB, then
maB maA .
If aA c aB, then aB c aA .
TRANSITIVE PROPERTY
Equality
Congruence
If AB CD and CD EF,
then AB EF.
If AB
& c CD
& and CD
& c EF
&,
then AB
& c EF
&.
If maA maB and
maB maC, then
maA maC .
If aA c aB and aB c aC,
then aA c aC .
Example 1
Properties of Equality and Congruence
Name the property that the statement illustrates.
a. DE DE
b. If aP c aQ and aQ c aR, then aP c aR.
Solution
a. Reflexive Property of Equality
b. Transitive Property of Congruence
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Geometry, Concepts and Skills Notetaking Guide • Chapter 2
Checkpoint Name the property that the statement illustrates.
1. If DF FG and FG GH, then DF GH.
Transitive Property of Equality
2. aP c aP
Reflexive Property of Congruence
3. If maS maT, then maT maS.
Symmetric Property of Equality
Example 2
Use Properties of Equality
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
M
N
P
Q
Definition of midpoint
Definition of midpoint
Transitive Property of Equality
Follow-Up
In Example 2, there is an explanation for each step in the
solution. In general, what can be used to explain steps in a
solution?
definitions, theorems, postulates, or properties
Lesson 2.6 • Geometry, Concepts and Skills Notetaking Guide
47
Checkpoint Complete the following exercise.
4. a1 and a2 are vertical angles and
a2 c a3. Show that a1 c a3.
a1 c a2
a2 c a3
a1 c a3
1
2
3
Vertical Angles Theorem
Given
Transitive Property of Congruence
ADDITION PROPERTY OF EQUALITY
Adding the same number to each side
of a true equation produces a true
equation.
x37
x337 3
SUBTRACTION PROPERTY OF EQUALITY
Subtracting the same number from
each side of a true equation produces
a true equation.
y 5 11
y 5 5 11 5
MULTIPLICATION PROPERTY OF EQUALITY
Multiplying each side of a true equation
by the same nonzero number produces
a true equation.
1
z 6
4
1
z p 4 6 p
4
4
DIVISION PROPERTY OF EQUALITY
Dividing each side of a true equation by
the same nonzero number produces a
true equation.
8x 16
8x 8 16 8
SUBSTITUTION PROPERTY OF EQUALITY
Substituting a number for a variable in a
true equation produces a true equation.
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Geometry, Concepts and Skills Notetaking Guide • Chapter 2
x7
2x 4 2( 7 ) 4
Example 3
Justify a Theorem
a1 and a2 are both supplementary
to a3. Show that a1 c a2.
1
2
3
Solution
ma1 ma3 180
Definition of supplementary angles
ma2 ma3 180
Definition of supplementary angles
ma1 ma3 ma2 ma3
Substitution Property of
Equality
ma1 ma2
Subtraction Property of Equality
a1 c a2
Definition of congruent
angles
Checkpoint Complete the following exercise.
5. In the diagram, M is the midpoint
of AB
&. Show that AB 2 p AM.
AM MB
A
M
Definition of midpoint
AM MB AB
Segment Addition Postulate
AM AM AB
Substitution Property of Equality
2 p AM AB
B
Distributive property
Lesson 2.6 • Geometry, Concepts and Skills Notetaking Guide
49