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

1) ΔCBF ≅ ΔCDF by (SSS)

2) ΔBFA ≅ ΔDFE by (SAS)

3) ΔCBE ≅ ΔCDA by (HL)

Step-by-step explanation:

1) Since BC ≅ DC and DF ≅ BF where CF ≅ CF (reflective property) we have;

ΔCBF ≅ ΔCDF by Side Side Side (SSS) rule of congruency

2) Since DF ≅ BF and FA ≅ FE where ∠DFE = ∠BFA (alternate angles)

Therefore;

ΔBFA ≅ ΔDFE by Side Angle Side (SAS) rule of congruency

3) Since FA ≅ FE and DF ≅ BF then where EB = FE + BF and AD = FA + DF

Where:

EB and AD are the hypotenuse sides of ΔCBE and ΔCDA respectively

We have that;

EB = AD from  FE + BF = FA + DF

Where we also have BC ≅ DC

Where:

BC and DC are the legs of ΔCBE and ΔCDA respectively

Then we have the following relation;

ΔCBE ≅ ΔCDA by Hypotenuse Leg (HL).