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 congruence postulates

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

Step2: Analyze \(\triangle CBF\) and \(\triangle CDF\)

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

Step3: Analyze \(\triangle BFA\) and \(\triangle DFE\)

We know \(FA\cong FE\), \(DF\cong BF\) and \(\angle BFA=\angle DFE\) (vertically opposite angles). By SAS postulate, \(\triangle BFA\cong\triangle DFE\).

Step4: Analyze \(\triangle CBE\) and \(\triangle CDA\)

Since \(\angle BCD = 90^{\circ}\), in right - triangles \(\triangle CBE\) and \(\triangle CDA\), \(BC\cong DC\) (given). But we don't have enough information to show hypotenuse and one leg are equal for HL.

Step5: Analyze \(\triangle CFD\) and \(\triangle EFD\)

We don't have enough information to show all three sides are equal for SSS.

Step6: Analyze \(\triangle ACF\) and \(\triangle ECF\)

We don't have enough information to show hypotenuse and one leg are equal for HL.

Answer:

\(\triangle CBF\cong\triangle CDF\) by SSS, \(\triangle BFA\cong\triangle DFE\) by SAS