Bot GPT 4-o mini To determine which relationships in the triangles are true, let's analyze each option provided based on the given information and congruences.
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△ACF ≅ △ECF by HL (Hypotenuse-Leg):
- Both triangles share the leg CF.
- Since ∠BCD is a right angle, we can also use the fact that AC and EC are hypotenuses in the right triangles ACF and ECF.
- Thus, this relationship is true.
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ΔCBF ≅ ΔCDF by SSS (Side-Side-Side):
- In triangles CBF and CDF:
- BC ≅ CD (given).
- BF ≅ DF (given).
- CF is common to both triangles.
- Therefore, this triangle relationship is true.
- In triangles CBF and CDF:
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ΔBFA ≅ ΔDFE by SAS (Side-Angle-Side):
- In triangles BFA and DFE:
- AF ≅ FE (given).
- BF ≅ DF (given).
- ∠BFA and ∠DFE are corresponding angles given that they are opposite to the equal sides and share a common vertex at F.
- Therefore, this statement is true.
- In triangles BFA and DFE:
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ΔCFD ≅ ΔEFD by SSS:
- In triangles CFD and EFD:
- CF is common.
- FD is equal to itself.
- However, without additional information, we can't confirm if CD and EF are congruent; both are not provided or deduced to be equal.
- Therefore, it is possible but not definitively true given the provided information.
- In triangles CFD and EFD:
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ΔCBE ≅ ΔCDA by HL:
- This triangle would require a right angle and a pair of equal sides to prove. From the information given, we do not have sufficient evidence to assume these triangles are congruent.
- Therefore, this statement cannot be confirmed.
Based on the analysis, the three relationships that are true are:
- △ACF ≅ △ECF by HL
- ΔCBF ≅ ΔCDF by SSS
- ΔBFA ≅ ΔDFE by SAS
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