The Transitive Property of Congruence
The Transitive Property
If you take a train from Belen to Albuquerque, and then continue on that train to Santa Fe, you have actually gone from Belen to Santa Fe. The transitive property is like this in the following sense: If you know one angle is congruent to another, say , and that other angle is congruent to a third angle, say, then you know the first angle is congruent to the third: . Basically, the transitive property tells us we can substitute a congruent angle with another congruent angle. This is really a property of congruence, and not just angles. If two segments are each congruent to a third segment, then they are congruent to each other, and if two triangles are congruent to a third triangle, then they are congruent to each other. So we can state the transitive property this way:
Transitive Property: If two geometric objects are congruent to a third geometric object, then they are congruent to each other.
Applications
Problem 1:
Given: 

Prove: is the midpoint of
Proof: By the transitive property, it follows that since both are congruent to . Therefore is the midpoint of since the midpoint of a segment splits it into two congruent pieces.
Problem 2:
Given: 

Prove: bisects
Proof: "Bisects" means "cuts in half," so we must show cuts into two equal angles. From the transitive property it follows that since they are both congruent to . Therefore bisects .
Problem 3:
Given:

,
lines L_{1} and L_{2} are parallel, lines L_{3} and L_{4} are parallel 

Prove:
Proof: Since L_{3} and L_{4} are parallel, , since they are alternate interior angles for the transversal L_{2}. Therefore by the transitive property. Since L_{1} and L_{2} are parallel, since they are corresponding angles for transversal L_{4}. Applying the transitive property again, we have .
Complements and Supplements
Two rather obvious results similar to the transitive property are these:
Theorem: Complements of congruent angles are congruent.
Proof: If two angles are congruent, then their measures are equal. Let us call the common measure a. Then a is a number between 0^{o} and 180^{o}. Their complements are (90 – a)^{o}, and so they are equal to. Therefore their complements are congruent.
Theorem: Supplements of supplementary angles are congruent.
The proof is essentially the same as for the previous theorem.
In addition, we can also state this rather obvious result:
The Reflexive Property of Congruence:
Any geometric object is congruent to itself.
Here are a couple of problems involving these concepts:
Problem 1:
Given: 
and are complements, and are complements 
Prove:
Proof: Since is congruent to itself (reflexive property), and are complements of congruent angles, so they are congruent.
Problem 2:
Given:  , and are supplements 
Prove:
Proof: and are supplements because they form a linear pair. Therefore (since and are supplements) . Since , it follows that by the transitive property.
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