How to Prove Angles Are Complementary or Supplementary
Using Theorems of Like Multiples and Like Divisions in Proofs
Proving Vertical Angles Are Congruent

The Transitive and Substitution Properties

You’re probably already familiar with the Transitive Property and the Substitution Property from algebra. If a = b and b = c, then a = c, right? That’s transitivity. And if a = b and b < c, then a < c. That’s substitution. Easy enough. Below, you see these theorems in greater detail:

  • Transitive Property (for three segments or angles): If two segments (or angles) are each congruent to a third segment (or angle), then they’re congruent to each other.


    The Transitive Property for three things is illustrated in the above figure.

  • Transitive Property (for four segments or angles): If two segments (or angles) are congruent to congruent segments (or angles), then they’re congruent to each other.


    The Transitive Property for four things is illustrated in the below figure.

  • Substitution Property: If two geometric objects (segments, angles, triangles, or whatever) are congruent and you have a statement involving one of them, you can pull the switcheroo and replace the one with the other. (Note that you will not be able to find the term “switcheroo” in your geometry glossary.)


    A figure isn’t especially helpful for this property, so one isn’t included here.

To avoid getting the Transitive and Substitution Properties mixed up, just follow these guidelines:

  • Use the Transitive Property as the reason in a proof when the statement on the same line involves congruent things.

  • Use the Substitution Property when the statement does not involve a congruence.

Check out this TGIF rectangle proof, which deals with angles:


–1 @ –2


No need for a game plan here because the proof is so short — take a look:

Statement 1:


Reason for statement 1: Given.

Statement 2:


Reason for statement 2: If two angles form a right angle, then they’re complementary (definition of complementary).

Statement 3:


Reason for statement 3: Given.

Statement 4:


Reason for statement 4: Substitution Property (statements 2 and 3; angle 2 replaces angle 1).

And for the final segment of the program, here’s a related proof, OSIM (Oh Shoot, It’s Monday):

image11.png image12.jpg

This is another incredibly short proof that doesn’t call for a game plan.

Statement 1:


Reason for statement 1: Given.


Reason for statement 2: A midpoint divides a segment into two congruent segments.


Reason for statement 3: Given.


Reason for statement 4: Transitive Property (for four segments; statements 2 and 3).

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Working with Definitions, Theorems, and Postulates
Using If-Then Logic
The Components of a Proof
Using Addition Theorems in Proofs
Using Subtraction Theorems in Proofs