9.05 Congruence transformations

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.

You may have been doing this already without even knowing it!

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?

We say objects are congruent if and only if one can be transformed into the other by a series of translations, rotation, and reflections.  We'll use this definition to prove that the reflexive, symmetric, and transitive properties of congruence are true.

Since we can perform a series of translations, rotations, and reflections that map an object onto itself (or bring it right back to where it started), the reflexive property holds for congruence.

If we can perform a series of translations, rotations, and reflections that bring object $A$A to object $B$B, then we can perform the reverse set of transformations to bring object $B$B to object $A$A.  This means that the order in a congruence statement doesn't matter, and the symmetric property of congruence holds.

Now suppose that object $A$A is congruent to object $B$B.  This means there is a series of transformations that map $A$A to $B$B.  Now suppose that object $B$B is congruent to object $C$C.  Then there's another series of transformations that map $B$B to $C$C.  If we combine the two series of transformations, then we can map $A$A to $C$C.  This means that $A$A and $C$C are also congruent.  Therefore, the transitive property of congruence holds.

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.

Now that we've established that the above properties of congruence are true, we can apply them to solve problems.  Let's try some examples together.

Practice questions

Question 1

Question 2

Question 3

We say that two triangles are congruent if one can be moved, rotated, and possibly reflected to lie on top of each other exactly. You can think of congruence as a more precise way of saying that two triangles are "the same". Here is an example to illustrate:

Here we are told by the markings on each triangle that the three sides are equal, and the three angles are equal as well - this is the given information. With a reflection, a rotation, and some translation, we will be able to place these triangles directly on top of each other.

Most of the time we are not given all three sides and all three angles. So the big question is: what is the minimum amount of information we need to know about two triangles to conclude that they are congruent? It turns out there are five different sets of information, all of which are sufficient to demonstrate congruence.

Side-side-side congruence (SSS)

If we know that each side of one triangle has a matching congruent side in the other triangle, then the triangles must be congruent. You can try this yourself with three straight objects; you can only make one triangle without changing the lengths, remembering that we are counting mirror images as being congruent as well:

This kind of congruence is called side-side-side, or just SSS.

 

Side-angle-side congruence (SAS)

If we know that two triangles have a pair of matching sides, and the angles between each pair are congruent, then the triangles must be congruent. You can try this yourself with any two straight objects - if you hold them together at a point and a certain angle apart along the rest of their lengths, there is only one triangle you can form by joining the ends together:

This kind of congruence is called side-angle-side, or just SAS.

 

Hypotenuse-leg congruence (HL)

This congruence test also uses two sides, and is an exception to the "the congruent angle must be between two congruent sides" rule. However, it only applies to right triangles. If two right triangles have congruent hypotenuses and one pair of legs are congruent, then the triangles must be congruent overall.

This congruence test is called hypotenuse-leg, or just HL.

Angle-side-angle and angle-angle-side congruences (ASA and AAS)

What if we only know that one side is congruent? In this case we need to know that there are two pairs of congruent angles. There are different cases that we need to treat separately, the first being when the two given angles lie on either end of the given side:

This kind of congruence is called angle-side-angle congruence, or just ASA.

The other way to use a single pair of congruent sides side to prove congruence of triangles is to know one pair of angles on the congruent sides are equal, and that the angles opposite the congruent sides are also equal:

This kind of congruence is called angle-angle-side congruence, or just AAS.

Here is a summary of the five triangle congruence tests.

 

Practice questions

QUESTION 4

QUESTION 5

QUESTION 6