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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?

RelationSymbolsExample
Equality=538=538
CongruenceMNPMNP
SimilarityMNPMNP
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?

RelationSymbolsExample
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?

RelationSymbolsExample
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.
ValueExample
Angle measurementsmA+mB=90°
Segment lengthsMN=PQ=5
AreaArea DEFG=81cm2
Ratio34=JKKL
We use congruence and similarity relations for geometric figures. We can't perform arithmetic operations like addition and multiplication on geometric figures.
FigureExample
AngleAC
Line segmentMNPQ
PolygonDEFGHI
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!

Want to join the conversation?

  • blobby green style avatar for user gn949968
    How do we know the difference between equality and congruence
    (27 votes)
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    • mr pink green style avatar for user David Severin
      numbers are equal to each other, and shapes are congruent to each other (same size and shape). Generally, if two angles, as examples, are congruent, then their measures are equal. If two quadrilaterals are congruent, the matching angles and matching sides would all have to be the same measure. Thus, shapes are congruent (because they usually on not directly on top of eache other) just means that matching pairs of sides and angles are all congruent.
      (28 votes)
  • male robot hal style avatar for user Fernandez Joaquin
    Why wont the voices stop
    (14 votes)
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  • female robot amelia style avatar for user syeboah
    I hope everyone is understanding his/her work well
    (10 votes)
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  • blobby green style avatar for user Ian Truong
    what is difference between equality and congruence in proofs
    (6 votes)
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    • duskpin tree style avatar for user WRaven
      Equality is when two things like varibles or measurements are the same numerically:
      21 / 3 = 9 - 2 because when you evaluate it they're the same number (7).

      Congruence is when to geometric figures like lines, angles, and polygons are the same geometrically:
      Triangle ABC is congruent to triangle XYZ because they have all the same side lengths and angle measures, which basically means they're the same shape.
      (11 votes)
  • blobby green style avatar for user ilinke.royse
    How do you know if the proof is true for EVERY triangle or just one specific triangle?
    (7 votes)
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  • hopper cool style avatar for user scrogginsna
    this is just a joke chat
    (5 votes)
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  • mr pants pink style avatar for user Sydney brewer
    In fields of green and brown,
    Amidst the trees and leaves,
    Lies a treasure hard to find,
    But oh, so worth the seek.

    It's the nut, so small and round,
    A source of protein, rich and sound,
    A snack to fill you up and more,
    A food that's hard to ignore.

    The walnut with its wrinkled shell,
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    The almond with its subtle taste,
    The pistachio that's hard to waste.

    These nuts are more than just a snack,
    They're symbols of a life well packed,
    Of strength, of health, of energy,
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    And when we crack them open wide,
    We find a treasure deep inside,
    A nut that's filled with all we crave,
    A food that's worth the time to save.

    So let us celebrate the nut,
    The humble food that's often cut,
    The source of all our energy,
    The key to living happily.
    (6 votes)
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  • blobby green style avatar for user ew3219991
    What is the difference between equality and congruency?
    (3 votes)
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    • leaf green style avatar for user kubleeka
      "Equal" means "these 'two' things are actually one thing with two names". We say 1+4=3+2 because '1+4' is a description of a certain number, and '3+2' is a description of the same exact number. Equality is a broad concept used across, and outside, all of math.

      Congruency is a concept specific to geometry. We say two figures are congruent if you can set one perfectly on top of the other without distorting either of them. More rigorously, if you can translate, rotate, and/or reflect one figure so that it lands perfectly on the other figure, then the two figures are congruent.

      If we say two triangles are congruent, it means they have the same side lengths, angles, and area, but they may be in different locations, tilted with respect to each other, or oriented differently. If we say triangle ABC is equal to triangle DEF, it means the triangles already fully coincide; maybe there is a single point that is named both A and D, a single point named both B and E, and a single point named both C and F.
      (8 votes)
  • blobby blue style avatar for user flora
    my brain cant process this :(
    (5 votes)
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  • aqualine ultimate style avatar for user Rooster
    please dont say that
    (4 votes)
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