Question

reorder the steps of the proof to make sure that steps that are logically dependent on prior steps are in the proper order.
given: \\( \overline { d e } \cong \overline { c e } \\) and \\( \overline { f e } \\) bisects \\( \angle d e c \\).
prove: \\( \overline { f a } \cong \overline { f b } \\).
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$$\begin{tabular}{|c|c|c|} \\hline step & statement & reason \\ \\hline 1 & \\( \\begin{array} { l } { \\overline { d e } \\cong \\overline { c e } } \\\\ { \\overline { f e } \\text { bisects } \\angle d e c } \\end{array} & \\text { given } \\\\ \\hline 2 & \\( \\overline { f a } \\cong \\overline { f b } & \\text { corresponding parts of congruent triangles are congruent (cpctc) } \\\\ \\hline 3 & \\( \\overline { f d } \\cong \\overline { f c } & \\text { corresponding parts of congruent triangles are congruent (cpctc) } \\\\ \\hline 4 & \\( \\triangle a c f \\cong \\triangle b d f & \\text { asa } \\\\ \\hline 5 & \\( \\angle d f c \\cong \\angle d f c & \\text { reflexive property } \\\\ \\hline 6 & \\( \\overline { f e } \\cong \\overline { f e } & \\text { reflexive property } \\\\ \\hline 7 & \\( \\angle f d e \\cong \\angle f c e & \\text { corresponding parts of congruent triangles are congruent (cpctc) } \\\\ \\hline 8 & \\( \\triangle f d e \\cong \\triangle f c e & \\text { sas } \\\\ \\hline 9 & \\( \\angle d e f \\cong \\angle c e f & \\text { an angle bisector divides an angle into two congruent angles } \\\\ \\hline \\end{tabular}$$

Explanation:

Step1: State given information

1. \( \overline{DE}\cong\overline{CE}\) (Given)
2. \( \overline{FE}\) bisects \( \angle DEC\) (Given)

Step2: Use angle - bisector property

1. \( \angle DEF\cong\angle CEF\) (An angle bisector divides an angle into two congruent angles)

Step3: Use reflexive property for side

1. \( \overline{FE}\cong\overline{FE}\) (Reflexive Property)

Step4: Prove \( \triangle FDE\cong\triangle FCE\)

1. \( \triangle FDE\cong\triangle FCE\) (SAS: \( \overline{DE}\cong\overline{CE}\), \( \angle DEF\cong\angle CEF\), \( \overline{FE}\cong\overline{FE}\))

Step5: Use CPCTC for \( \angle FDE\) and \( \angle FCE\)

1. \( \angle FDE\cong\angle FCE\) (Corresponding Parts of Congruent Triangles are Congruent (CPCTC))

Step6: Use CPCTC for \( \overline{FD}\) and \( \overline{FC}\)

1. \( \overline{FD}\cong\overline{FC}\) (Corresponding Parts of Congruent Triangles are Congruent (CPCTC))

Step7: Use reflexive property for angle

1. \( \angle DFC\cong\angle DFC\) (Reflexive Property)

Step8: Prove \( \triangle ACF\cong\triangle BDF\)

1. \( \triangle ACF\cong\triangle BDF\) (ASA: \( \angle FDE\cong\angle FCE\), \( \overline{FD}\cong\overline{FC}\), \( \angle DFC\cong\angle DFC\))

Step9: Use CPCTC for \( \overline{FA}\) and \( \overline{FB}\)

1. \( \overline{FA}\cong\overline{FB}\) (Corresponding Parts of Congruent Triangles are Congruent (CPCTC))

Answer:

The correct order of steps is: 1, 9, 6, 8, 7, 3, 5, 4, 2.