Properties of Segment Congruence

Congruence of segments follows the same reflexive, symmetric, and transitive patterns as equality. These three properties, combined with the definition of congruent segments, drive most proofs comparing lengths.

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Congruence of segments behaves in three predictable ways, mirroring the properties of equality you already know from algebra. These three properties — reflexive, symmetric, and transitive — appear in almost every geometric proof that compares lengths.

Concept Reflexive property

Every segment is congruent to itself. This sounds almost too obvious to state, but it is exactly the tool needed whenever a proof requires comparing a shared segment to itself across two figures.

Concept Symmetric property

The order in which a congruence is written does not matter. Whichever segment you happen to write first, the statement can always be flipped without changing its meaning.

Concept Transitive property

AB CD EF
is the link segment — it is congruent to both of the others, so those other two must also be congruent to each other.
Example Proving the transitive property

Given: and . Prove that .

Statement Reason
AB ≅ CD, CD ≅ EF Given
AB = CD, CD = EF Definition of congruent segments
AB = EF Transitive property of equality
AB ≅ EF Definition of congruent segments

The proof relies on switching between congruence and equality: the definition of congruent segments converts into , the transitive property of equality (from algebra) does the actual work, and the definition converts back to at the end.

Example Applying all three properties
1. Is true? — Yes, by the reflexive property; every segment is congruent to itself.
2. If , what can you say about and ? — By the symmetric property, .
3. If and , what is the relationship between and ? — By the transitive property, using as the link, .
Summary
  1. Reflexive property: a segment is always congruent to itself, AB ≅ AB.
  2. Symmetric property: if AB ≅ CD, then CD ≅ AB — the order can be reversed.
  3. Transitive property: if AB ≅ CD and CD ≅ EF, then AB ≅ EF — the shared middle segment links the two outer ones.
  4. Proofs using these properties typically switch to equality via the definition of congruent segments, apply the matching algebraic property, then switch back.
283. Relations and Functions
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