If △ ABC ≅ △ EFG and △ EFG ≅ △ JKL, then △ ABC≅ △ JKL.
By definition, two polygons are congruent when their corresponding parts are congruent. By following the order of the letters, we can list the corresponding parts in a table.
△ ABC ≅ △ EFG
△ EFG ≅ △ JKL
∠A≅ ∠E
∠E≅ ∠J
∠B≅ ∠F
∠F≅ ∠K
∠C ≅ ∠G
∠G ≅ ∠L
A B ≅ E F
E F ≅ J K
B C ≅ F G
F G ≅ K L
A C ≅ E G
E G ≅ J L
By applying the Transitive Property of Congruence on each row of the table, we obtain the following congruence relations.
ccc
∠A≅ ∠J & & A B ≅ J K
∠B≅ ∠K & and & B C ≅ K L
∠C ≅ ∠L & & A C ≅ J L
Next, by the definition of congruent polygons we conclude that △ A B C ≅ △ J K L.
Completed Proof
Given: & △ ABC ≅ △ EFG and △ EFG ≅ △ JKL
Prove: & △ ABC≅ △ JKL
Proof: Since △ ABC ≅ △ EFG and by the definition of congruent polygons, we get ∠A ≅ ∠E, ∠B ≅ ∠F, ∠C ≅ ∠G, AB ≅ EF, BC ≅ FG, and AC ≅ EG. Similarly, since △ EFG ≅ △ JKL we obtain ∠E ≅ ∠J, ∠F ≅ ∠K, ∠G ≅ ∠L, EF ≅ JK, FG ≅ KL, and EG ≅ JL.
Next, by using the Transitive Property of Congruence we get that ∠A ≅ ∠J, ∠B ≅ ∠K, ∠C ≅ ∠L, AB ≅ JK, BC ≅ KL, AC ≅ JL. Finally, by the definition of congruent polygons, we conclude that △ ABC≅ △ JKL.
Extra
The Transitive Property of Congruence In a Fun Way
For understanding, outside of a strict math concept, let's consider the Transitive Property of Congruence using relationships we have with our friends!
If Julio's headphones are the same size as Michelle's headphones, and Bianca's headphones are the same size as Michelle's headphones, then Julio and Bianca's headphones are the same size.
