6.04 Properties of equality and congruence

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.	$\angle ABC\cong\angle ABC$∠ABC≅∠ABC
Symmetric property	The reverse congruence statement is also true.	If $\angle ABC\cong\angle DEF$∠ABC≅∠DEF then $\angle DEF\cong\angle ABC$∠DEF≅∠ABC.
Transitive property	Congruence is transferable.	If $\angle ABC\cong\angle DEF$∠ABC≅∠DEF and $\angle DEF\cong\angle GHI$∠DEF≅∠GHI, then $\angle ABC\cong\angle GHI$∠ABC≅∠GHI.

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

Question 2

Question 3