Question

given: ∠bcd is right, bc≅dc; df≅bf; fa≅fe. which relationships in the diagram are true? select three options. △acf≅△ecf by hl △cbf≅△cdf by sss △bfa≅△dfe by sas △cfd≅△efd by sss △cbe≅△cda by hl

Explanation:

Step1: Recall triangle - congruence postulates

HL (Hypotenuse - Leg) is for right - triangles where the hypotenuse and one leg are congruent. SSS (Side - Side - Side) is when all three sides of two triangles are congruent. SAS (Side - Angle - Side) is when two sides and the included angle of two triangles are congruent.

Step2: Analyze \(\triangle ACF\) and \(\triangle ECF\)

We know that \(\angle BCD\) is right, \(BC\cong DC\), \(DF\cong BF\), \(FA\cong FE\). In \(\triangle ACF\) and \(\triangle ECF\), \(FA = FE\) (given), \(CF=CF\) (common side), and since \(\angle BCD = 90^{\circ}\), we can consider right - triangles. But we don't have enough information to use HL.

Step3: Analyze \(\triangle CBF\) and \(\triangle CDF\)

We have \(BC\cong DC\), \(DF\cong BF\), and \(CF = CF\) (common side). By SSS, \(\triangle CBF\cong\triangle CDF\).

Step4: Analyze other options

For \(\triangle CBE\) and \(\triangle CDA\), we don't have enough information to use HL. For \(\triangle CFD\) and \(\triangle EFD\), we don't have enough information to use SSS. For \(\triangle BFA\) and \(\triangle DFE\), we don't have enough information to use SAS.

Answer:

\(\triangle CBF\cong\triangle CDF\) by SSS