To make our process clear, we will develop the proof of each property one at a time. Then, at the end of each part, we will summarize the proof in a two-column table, making things even more clear.
Reflexive Property of Similarity
Let's consider â–³ ABC. When writing the following two relations, use the fact that the congruence relation is reflexive. That fact can be presented in the following manner.
∠A ≅ ∠A and ∠B ≅ ∠B
Then, by the Angle-Angle (AA) Similarity Postulate, we can conclude that â–³ ABC ~ â–³ ABC. This proves the Reflexive Property of Similarity.
Statements
Reasons
1.
â–³ ABC
1.
Given
2.
∠A≅ ∠A and ∠B ≅ ∠B
2.
Reflexive property of congruence
3.
â–³ ABC ~ â–³ ABC
3.
Angle-Angle (AA) Similarity Postulate
Symmetric Property of Similarity
If â–³ ABC~ â–³ DEF, then the corresponding angles are congruent by definition.
∠A ≅ ∠D and ∠B ≅ ∠E
The Symmetric Property of Congruence allows us to rewrite the relations above as follows.
∠D ≅ ∠A and ∠E ≅ ∠B
Again, by applying the AA Similarity Postulate, we conclude that â–³ DEF~â–³ ABC, which proves the Symmetric Property of Similarity.
Statements
Reasons
1.
â–³ ABC~â–³ DEF
1.
Given
2.
∠A≅ ∠D and ∠B ≅ ∠E
2.
Definition of similar triangles
3.
∠D≅ ∠A and ∠E ≅ ∠B
3.
Symmetric property of congruence
4.
â–³ DEF ~ â–³ ABC
4.
Angle-Angle (AA) Similarity Postulate
Transitive Property of Similarity
If â–³ ABC~ â–³ DEF and â–³ DEF~ â–³ GHI, then the corresponding angles are congruent by definition.
∠A ≅ ∠D and ∠D ≅ ∠G
∠B ≅ ∠E and ∠E ≅ ∠H
Applying the Transitive Property of Congruence, we get the two relations below.
∠A ≅ ∠G and ∠B ≅ ∠H
Consequently, â–³ ABC~â–³ GHI thanks to the AA Similarity Postulate. This proves the Transitive Property of Similarity.
Statements
Reasons
1.
â–³ ABC~â–³ DEF and â–³ DEF ~ â–³ GHI
1.
Given
2.
∠A≅ ∠D and ∠B ≅ ∠E ∠D ≅ ∠G, and ∠E ≅ ∠H
