Question

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 { a e } \parallel \overline { f d } \\), \\( \overline { e c } \cong \overline { b f } \\) and \\( \angle a b f \cong \angle e c d \\).
prove: \\( \overline { a b } \cong \overline { c d } \\).

step	statement	reason
2	\\( \overline { b c } \cong \overline { b c } \\)	reflexive property
3	\\( \angle e c d \\) and \\( \angle e c b \\) are supplementary	if two angles form a linear pair, then they are supplementary
4	\\( \angle a b f \\) and \\( \angle f b d \\) are supplementary	if two angles form a linear pair, then they are supplementary
5	\\( \triangle a e c \cong \triangle d f b \\)	aas
6	\\( \angle a \cong \angle d \\)	parallel lines cut by a transversal form congruent alternate interior angles
7	\\( \overline { a c } \cong \overline { b d } \\)	corresponding parts of congruent triangles are congruent (cpctc)
8	\\( \angle f b d \cong \angle e c b \\)	if two angles are supplements of the same angle (or congruent angles), then they are congruent
9	\\( \overline { a b } \cong \overline { c d } \\)	congruent segments subtracted from congruent segments form congruent segments

Explanation:

Step1: State given information

$\overline{AE}\parallel\overline{FD}$, $\overline{EC}\cong\overline{BF}$, $\angle ABF\cong\angle ECD$ (Given)

Step2: Use parallel line property

$\angle A\cong\angle D$ (Parallel lines cut by a transversal form congruent alternate interior angles)

Step3: Use reflexive property

$\overline{BC}\cong\overline{BC}$ (Reflexive Property)

Step4: Use linear - pair and supplementary - angle property

$\angle ECD$ and $\angle ECB$ are supplementary (If two angles form a linear pair, then they are supplementary)
$\angle ABF$ and $\angle FBD$ are supplementary (If two angles form a linear pair, then they are supplementary)
$\angle FBD\cong\angle ECB$ (If two angles are supplements of the same angle (or congruent angles), then they are congruent)

Step5: Prove triangle congruence

$\triangle AEC\cong\triangle DFB$ (AAS: $\angle A\cong\angle D$, $\overline{EC}\cong\overline{BF}$, $\angle ECB\cong\angle FBD$)

Step6: Use CPCTC

$\overline{AC}\cong\overline{BD}$ (Corresponding Parts of Congruent Triangles are Congruent (CPCTC))

Step7: Prove segment congruence

$\overline{AB}\cong\overline{CD}$ (Congruent segments subtracted from congruent segments form congruent segments: $\overline{AC}-\overline{BC}=\overline{AB}$, $\overline{BD}-\overline{BC}=\overline{CD}$)

Answer:

The correct order of steps is: 1 (Given $\overline{AE}\parallel\overline{FD}$, $\overline{EC}\cong\overline{BF}$, $\angle ABF\cong\angle ECD$), 6 ($\angle A\cong\angle D$), 2 ($\overline{BC}\cong\overline{BC}$), 3 ($\angle ECD$ and $\angle ECB$ are supplementary), 4 ($\angle ABF$ and $\angle FBD$ are supplementary), 8 ($\angle FBD\cong\angle ECB$), 5 ($\triangle AEC\cong\triangle DFB$), 7 ($\overline{AC}\cong\overline{BD}$), 9 ($\overline{AB}\cong\overline{CD}$)