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In two triangles ABC and DEF, if AB=EF, BC=DF and CA=DE, then:
ΔABC\cong\triangle DEF
ΔDEF\cong\triangle BCA
ΔBAC\cong\triangle FDE
ΔCBA\cong\triangle DFE
ΔCBA≅ΔDFE
Match the corresponding vertices by matching the equal side lengths: AB corresponds to EF (vertices A,B correspond to E,F in order or F,E reversed), BC corresponds to DF (vertices B,C correspond to D,F), and CA corresponds to DE (vertices C,A correspond to D,E). Tracing the vertices in order gives ΔCBA \cong ΔDFE.
Two triangles ABC and DEF with corresponding side lengths equal: \bar{AB} = \bar{EF}, \bar{BC} = \bar{DF}, and \bar{CA} = \bar{DE}.
ΔABC≅ΔFDE
Match the corresponding vertices by matching the equal side lengths: AB corresponds to EF (vertices A,B correspond to E,F in order or F,E reversed), BC corresponds to DF (vertices B,C correspond to D,F), and CA corresponds to DE (vertices C,A correspond to D,E). Tracing the vertices in order gives ΔCBA \cong ΔDFE.
Assuming the triangles are congruent in alphabetical order (ΔABC \cong ΔDEF) without checking the given side equality mappings.
Analyze Given Side Relations
We are given three equality relations between the sides of triangles ABC and DEF: AB=EF, BC=DF, and CA=DE.
AB=EF,BC=DF,CA=DE
Identify Corresponding Vertices
From the equality of sides, vertex A corresponds to vertex F (since AB=EF and CA=DE, vertex A is common to sides AB and CA, which correspond to sides EF and DE having vertex E in common? Let's check: AB = EF, BC = DF, CA = DE. Thus: A corresponds to F, B corresponds to E, and C corresponds to D.
A↔F,B↔E,C↔D
Formulate Congruence Statement
Listing the corresponding vertices in order: vertex C corresponds to D, vertex B corresponds to E, vertex A corresponds to F. Therefore, the congruent triangles statement is ΔCBA≅ΔDFE.
ΔCBA≅ΔDFE
D is correct because matching the corresponding sides AB=EF, BC=DF, and CA=DE establishes the vertex correspondence C↔D, B↔F, and A↔E, which yields ΔCBA≅ΔDFE.
Always write congruent triangles by properly aligning the corresponding vertices in the same order as their equal sides.
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