Asked by neko :3
Given: ∠BCD is right; BC ≅ DC; DF ≅ BF; FA ≅ FE
Triangles A C D and E C B overlap and intersect at point F. Point B of triangle E C B is on side A C of triangle A C D. Point D of triangle A C D is on side C E of triangle E C D. Line segments B C and C D are congruent. Line segments B F and F D are congruent. Line segments A F and F E are congruent.
Which relationships in the diagram are true? Select three options.
△ACF ≅ △ECF by HL
ΔCBF ≅ ΔCDF by SSS
ΔBFA ≅ ΔDFE by SAS
ΔCFD ≅ ΔEFD by SSS
ΔCBE ≅ ΔCDA by HL
Triangles A C D and E C B overlap and intersect at point F. Point B of triangle E C B is on side A C of triangle A C D. Point D of triangle A C D is on side C E of triangle E C D. Line segments B C and C D are congruent. Line segments B F and F D are congruent. Line segments A F and F E are congruent.
Which relationships in the diagram are true? Select three options.
△ACF ≅ △ECF by HL
ΔCBF ≅ ΔCDF by SSS
ΔBFA ≅ ΔDFE by SAS
ΔCFD ≅ ΔEFD by SSS
ΔCBE ≅ ΔCDA by HL
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To determine which relationships in the triangles are true, let's analyze each option provided based on the given information and congruences.
1. **△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.
2. **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.
3. **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.
4. **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.
5. **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:
1. △ACF ≅ △ECF by HL
2. ΔCBF ≅ ΔCDF by SSS
3. ΔBFA ≅ ΔDFE by SAS
1. **△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.
2. **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.
3. **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.
4. **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.
5. **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:
1. △ACF ≅ △ECF by HL
2. ΔCBF ≅ ΔCDF by SSS
3. ΔBFA ≅ ΔDFE by SAS
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