# 2.04 Properties of equality and congruence

Lesson

Just as we can apply the properties of equality to real numbers to solve for an equation, we can also apply them to real number measurements when solving for values in a geometric diagram.

### Properties of equality for real numbers

What exactly are the properties of equality?  Put simply, they're assumptions of how we can manipulate equations while maintaining truth.  They're the foundation for most of mathematics.

The table below summarizes the properties and gives an example of how we might apply them in geometry using segment lengths.

Property of equality Meaning Example in geometry
Reflexive property Anything is equal to itself. $AB=AB$AB=AB
Symmetric property The reverse equation is also true. If $AB=CD$AB=CD then $CD=AB$CD=AB.
Transitive property Equality is transferable. If $AB=CD$AB=CD and $CD=EF$CD=EF, then $AB=EF$AB=EF.
Addition / Subtraction property Equals added to (or subtracted from) equals are still equal. If $AB=CD$AB=CD then $AB+5=CD+5$AB+5=CD+5
Multiplication / Division property Equals multiplied (or divided) by equals are still equal, as long as the division isn't by $0$0. If $AB=CD$AB=CD then $\frac{AB}{5}=\frac{CD}{5}$AB5=CD5

Since these properties are true for any real number, they're also true for real number measurements, such as segment lengths, or angle measures.

### Properties of congruence

We've established that measurements of geometrical objects follow the same properties of equality that real numbers do.  What can we say about the geometrical objects themselves?

Because segments with the same length are congruent, the congruence of segments is also reflexive, symmetric, and transitive.  The same is true for congruent angles.

The table below summarizes the properties of congruence with some statements using angles as examples.

Property of congruence Meaning Example in geometry
Reflexive property Anything is congruent to itself. $\angle ABC\cong\angle ABC$ABCABC
Symmetric property The reverse congruence statement is also true. If $\angle ABC\cong\angle DEF$ABCDEF then $\angle DEF\cong\angle ABC$DEFABC.
Transitive property Congruence is transferable. If $\angle ABC\cong\angle DEF$ABCDEF and $\angle DEF\cong\angle GHI$DEFGHI, then $\angle ABC\cong\angle GHI$ABCGHI.

Note: it's possible to show that the above properties are true using the definition of congruent segments or angles and the properties of equality.  Try writing a paragraph or two-column proof that proves the properties.

#### Practice questions

##### Question 1

Consider the true expression:

If $AB=CD$AB=CD then $CD=AB$CD=AB

Which of the following properties does this illustrate?

1. The transitive property of equality.

A

The multiplication property of equality.

B

The reflexive property of equality.

C

D

The symmetric property of equality.

E

The transitive property of equality.

A

The multiplication property of equality.

B

The reflexive property of equality.

C

D

The symmetric property of equality.

E

##### Question 2

Fill in the blank so that the resulting statement is true.

1. The symmetric property of equality states that if $AB=CD$AB=CD then $CD$CD = $\editable{}$.

##### Question 3

Consider the following true statement:

$AB+CD=32$AB+CD=32

1. Which of the following is a valid deduction from this statement using only a single property of equality?

$BA+DC=16+16$BA+DC=16+16

A

$AB+AB=32+CD$AB+AB=32+CD

B

$2AB=64-CD$2AB=64CD

C

$32=AB+CD$32=AB+CD

D

$BA+DC=16+16$BA+DC=16+16

A

$AB+AB=32+CD$AB+AB=32+CD

B

$2AB=64-CD$2AB=64CD

C

$32=AB+CD$32=AB+CD

D