Question

using triangle congruence theorems
given: \\( \angle b c d \\) is right; \\( \overline{b c} \cong \overline{d c} ; \overline{d f} \cong \overline{b f} ; \overline{f a} \cong \overline{f e} \\)
which relationships in the diagram are true? choose three correct answers.
\\( \triangle c b e \cong \triangle c d a \\) by hl
\\( \triangle a c f \cong \triangle e c f \\) by hl
\\( \triangle b f a \cong \triangle d f e \\) by sas
\\( \triangle c f d \cong \triangle e f d \\) by sss
\\( \triangle c b f \cong \triangle c d f \\) by sss

Explanation:

Step1: Analyze $\triangle CBE\cong\triangle CDA$ by HL

Since $\angle BCD$ is right, $\angle CBE=\angle CDA = 90^{\circ}$ (supplementary to right angle). $BC\cong DC$ (given), $BE=BF + FE$, $DA = DF+FA$. As $DF\cong BF$ and $FA\cong FE$, $BE\cong DA$. By HL (Hypotenuse - Leg) theorem, $\triangle CBE\cong\triangle CDA$.

Step2: Analyze $\triangle ACF\cong\triangle ECF$ by HL

$FA\cong FE$ (given), $CF$ is common side. $\angle CFA=\angle CFE = 90^{\circ}$ (vertical angles are equal and can be shown to be right angles from the congruent segments and triangle properties). By HL theorem, $\triangle ACF\cong\triangle ECF$.

Step3: Analyze $\triangle CBF\cong\triangle CDF$ by SSS

$BC\cong DC$ (given), $BF\cong DF$ (given), $CF$ is common side. By SSS (Side - Side - Side) theorem, $\triangle CBF\cong\triangle CDF$.

Step4: Analyze $\triangle BFA\cong\triangle DFE$ by SAS

$BF\cong DF$ (given), $FA\cong FE$ (given), $\angle BFA=\angle DFE$ (vertical angles). By SAS (Side - Angle - Side) theorem, $\triangle BFA\cong\triangle DFE$.

Step5: Analyze $\triangle CFD\cong\triangle EFD$ by SSS

$CF
eq EF$ (from $\triangle ACF\cong\triangle ECF$, $AC = EC$ but $CF$ is a leg, not equal to $EF$ in general from given information). So $\triangle CFD\cong\triangle EFD$ by SSS is false.

Answer:

$\triangle CBE\cong\triangle CDA$ by $HL$, $\triangle ACF\cong\triangle ECF$ by $HL$, $\triangle BFA\cong\triangle DFE$ by $SAS$, $\triangle CBF\cong\triangle CDF$ by $SSS$