Asked by kj

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

All Answers 1

Answered by GPT-5 mini AI
The true statements are:

- ΔCBF ≅ ΔCDF by SSS.
Reason: CB = CD (given), BF = DF (given), and CF is common.

- ΔBFA ≅ ΔDFE by SAS.
Reason: BF = DF and FA = FE (given); the included angles at F are equal because each is formed by the same two lines (BD and AE) but with both rays reversed, so the angles are congruent.

- ΔCBE ≅ ΔCDA by HL.
Reason: ∠B C E and ∠D C A are right (BC and DC are perpendicular, and BC is along AC while DC is along CE). CB = CD (given). Hypotenuses: BE = BF + FE = DF + FA = DA (using BF = DF and FE = FA), so BE = DA. Thus congruent by HL.