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Properties of congruence and equality

Learn when to apply the reflexive property, transitive, and symmetric properties in geometric proofs. Learn the relationship between equal measures and congruent figures.
There are lots of ways to write proofs, and some are more formal than others. In very formal proofs, we justify statements that may feel obvious to you. The reason we justify them is that those claims only work with certain types of relations. What's true with the equality relation isn't necessarily true with the inequality relation, for example.
Let's look at some of these properties. We'll use the symbol to represent an unknown relation.

Reflexive property

When a relation has a reflexive property, it means that the relation is always true between a thing and itself. So AA.

What are some relations that use it?

We use the reflexive property a lot when we're looking at shapes that share sides or angles.
Two triangles M N Q and P N Q share the same side N Q. Side N M is congruent to side N P. Side M Q is congruent to side P Q. Side N Q is congruent to itself.
If we were talking about how MNQ and PNQ relate, we might state that NQNQ because of the reflexive property.

What are some relations that don't?

Strict inequalities don't have a reflexive property. For example, 33.
Being somebody's mother isn't a reflexive relationship. I am not my own mother.

Symmetric property

When a relation has a symmetric property, it means that the if relation is true between two things, it is true in either order. If AB, then BA.

What are some relations that use it?

Equality=If 8=113, then 113=8.
CongruenceIf VWXY, then XYVW.
SimilarityIf ABCDLMNP, then LMNPABCD.
ParallelismIf line m line n, then line n line m.
PerpendicularityIf STUV, then UVST.
By most people's definitions, friendship is a symmetric relationship. If Alaia is friends with Kolton, then Kolton is friends with Alaia.

What are some relations that don't?

Strict inequalities don't have a symmetric property. For example, 10<100, but 10010.
Being somebody's mother also isn't a symmetric relationship. If Karin is Santino's mother, then Santino cannot be Karin's mother.

Transitive property

When a relation has a transitive property, then two things that relate to a common middle thing also relate to each other. If AB and BC, then AC.

What are some relations that use it?

Equality=If mF=mG and mG=mH, then mF=mH.
CongruenceIf RSTWXY and WXYFGH, then RSTFGH.
SimilarityIf circle A circle B and circle B circle D, then circle A circle D.
ParallelismIf JKLM and LMNO, then JKNO.

What are some relations that don't?

Perpendicularity is not transitive.
3 lines. Line AC is perpendicular to line AB. Line AC is also perpendicular to line CD.
In the figure, ABAC and ACCD, but AB is parallel to, not perpendicular to, CD.
Friendship is also not transitive. If Ezekiel is friends with Romina, and Romina is friends with Nash, we don't know whether or not Ezekiel is friends with Nash.

Equality versus congruence

Equality and congruence are closely connected, but different. We use equality relations for anything we can express with numbers, including measurements, scale factors, and ratios.
Angle measurementsmA+mB=90°
Segment lengthsMN=PQ=5
AreaArea DEFG=81cm2
We use congruence and similarity relations for geometric figures. We can't perform arithmetic operations like addition and multiplication on geometric figures.
Line segmentMNPQ
CircleAll circles are similar to all other circles.
There are three very useful theorems that connect equality and congruence.
So in the following figure, we're given that AB=CD=3.2.
Quadrilateral ABCD with sides AB and CD labeled 3.2, and sides BC and DA labeled 5.
In a very formal proof, we would need a separate line to claim ABCD. More casual proofs use equal measures and congruent parts interchangeably. Check with your class to see which you need!

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