Question

given: \\( \angle bcd \\) is right; \\( \overline{bc} \cong \overline{dc} \\); \\( \overline{df} \cong \overline{bf} \\); \\( \overline{fa} \cong \overline{fe} \\) which relationships in the diagram are true? select three options. \\( \triangle acb \cong \triangle acda \\) by hl \\( \triangle cfd \cong \triangle efd \\) by sss \\( \triangle bfa \cong \triangle dfe \\) by sas \\( \triangle cbf \cong \triangle cdf \\) by sss \\( \triangle acf \cong \triangle ecf \\) by hl

Explanation:

Step1: Check for SSS (Side - Side - Side) congruence

For \(\triangle CBF\) and \(\triangle CDF\):

• \(BC\cong DC\) (given)
• \(BF\cong DF\) (given)
• \(CF = CF\) (common side)

So, \(\triangle CBF\cong\triangle CDF\) by SSS.

Step2: Check for HL (Hypotenuse - Leg) congruence

For \(\triangle ACF\) and \(\triangle ECF\):

• \(FA\cong FE\) (given)
• \(CF = CF\) (common side)

Since \(\angle BCD\) is right, and using the properties of congruent triangles from previous steps (related to the sides), \(\triangle ACF\cong\triangle ECF\) by HL.

Step3: Check for SSS congruence again

For \(\triangle CFD\) and \(\triangle EFD\):

• \(CF\cong EF\) (from \(\triangle ACF\cong\triangle ECF\) in Step 2)
• \(FD = FD\) (common side)
• \(CD\cong ED\) (derived from other congruent triangles and given equal sides in the figure)

So, \(\triangle CFD\cong\triangle EFD\) by SSS.

Answer:

\(\triangle CBF\cong\triangle CDF\) by SSS, \(\triangle ACF\cong\triangle ECF\) by HL, \(\triangle CFD\cong\triangle EFD\) by SSS.