Question

given: \\(\overline{bf} \cong \overline{cf}\\) and \\(\angle adf \cong \angle aef\\).
prove: \\(\overline{dc} \cong \overline{eb}\\).

step	statement	reason
2	\\(\angle adf \cong \angle aef\\)	given
3	\\(\angle bdf\\) and \\(\angle adf\\) are supplementary	if two angles form a linear pair, then they are supplementary
4	\\(\angle cef\\) and \\(\angle aef\\) are supplementary	if two angles form a linear pair, then they are supplementary
5	\\(\angle bdf \cong \angle cef\\)	if two angles are supplements of the same angle (or congruent angles), then they are congruent
6	\\(\angle bfd \cong \angle cfe\\)	vertical angles are congruent
7	\\(\triangle bfd \cong \triangle cfe\\)	aas
8	\\(\overline{df} \cong \overline{ef}\\)	corresponding parts of congruent triangles are congruent (cpctc)
9	\\(\overline{dc} \cong \overline{eb}\\)	in a triangle, sides opposite of congruent angles are congruent (and other segment congruence reasons from the list: corresponding parts of congruent triangles are congruent (cpctc), a segment bisector divides a segment into two congruent segments, a midpoint divides a segment into two congruent segments, opposite sides of a parallelogram/rhombus/rectangle/square are congruent, the diagonals of a rectangle/square are congruent, all sides of a rhombus/square are congruent, congruent segments added to congruent segments form congruent segments, congruent segments subtracted from congruent segments form congruent segments, segments that are half the lengths of congruent segments are congruent, segments that are twice the lengths of congruent segments are congruent)

Explanation:

Step1: Analyze Step 2 Reason

We know that \(\angle ADF\cong\angle AEF\) (given). \(\angle BDF\) and \(\angle ADF\) form a linear pair, and \(\angle CEF\) and \(\angle AEF\) form a linear pair. The reason for \(\angle BDF\) and \(\angle ADF\) being supplementary (and same for \(\angle CEF\) and \(\angle AEF\)) is "If two angles form a linear pair, then they are supplementary". So for Step 2, the reason is that linear - pair angles are supplementary.

Step2: Analyze Step 3 Reason

We have \(\angle BDF\) supplementary to \(\angle ADF\) and \(\angle CEF\) supplementary to \(\angle AEF\), and \(\angle ADF\cong\angle AEF\). By the theorem "If two angles are supplements of the same angle (or congruent angles), then they are congruent", we can conclude that \(\angle BDF\cong\angle CEF\). So the reason for Step 3 is this supplement - of - congruent - angles theorem.

Step3: Analyze Step 4 Reason

\(\angle BFD\) and \(\angle CFE\) are vertical angles. The reason for \(\angle BFD\cong\angle CFE\) is "Vertical angles are congruent".

Step4: Analyze Step 5 Reason

We know that \(BF\cong CF\) (given), \(\angle BDF\cong\angle CEF\) (from Step 3), and \(\angle BFD\cong\angle CFE\) (from Step 4). By the AAS (Angle - Angle - Side) congruence criterion, we can conclude that \(\triangle BFD\cong\triangle CFE\). So the reason for Step 5 is AAS.

Step5: Analyze Step 6 Reason

Since \(\triangle BFD\cong\triangle CFE\), by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, we have \(DF\cong EF\). So the reason for Step 6 is CPCTC.

Step6: Analyze Step 7 Reason

We know that \(DF\cong EF\) (from Step 6). Also, from the given and previous steps, we can consider the triangle properties. In a triangle, sides opposite congruent angles are congruent. But more directly, since we want to prove \(DC\cong EB\), and we can use the fact that we have congruent triangles and segment relationships. But actually, since \(\triangle BFD\cong\triangle CFE\) and with the other relationships, and then using the properties of triangles (sides opposite congruent angles or CPCTC - like for the other triangle), but more precisely, after getting \(DF = EF\) and using the given \(BF = CF\) and the angle relationships, and then by the theorem "In a triangle, sides opposite of congruent angles are congruent" or by CPCTC for the final triangle, we can get \(DC\cong EB\). But actually, since we have \(DF\cong EF\) and the other congruences, and then by the triangle congruence and side - angle - side or other, but the key is that after establishing the congruence of triangles and the equality of segments, we use the property of triangles (sides opposite congruent angles) or CPCTC - extended. But the reason for \(DC\cong EB\) is that in the relevant triangle (after all the congruences), sides opposite congruent angles are congruent (or by CPCTC for the larger triangle relationship). But more accurately, since we have \(\triangle BFD\cong\triangle CFE\), and then we can show that \(\triangle DFC\cong\triangle EFB\) (by SAS, since \(DF = EF\), \(CF = BF\), and \(\angle DFC=\angle EFB\) (vertical angles or from previous congruences)), and then by CPCTC, \(DC\cong EB\). But the main reason for Step 7 (wait, Step 7 is \(DF\cong EF\) with reason CPCTC, Step 8 is \(DC\cong EB\)): The reason for Step 8 is "Corresponding Parts of Congruent Triangles are Congruent" (CPCTC) or "In a triangle, sides opposite of congruent angles are congruent" (if we consider the triangle with \(DF = EF\) and the other angles). But since we have congruent triangles from the previous steps, the reason for \(DC\cong EB\) is CPCTC (after showing the relevant triangle congruence).

Answer:

1. Step 2 Reason: If two angles form a linear pair, then they are supplementary.
2. Step 3 Reason: If two angles are supplements of the same angle (or congruent angles), then they are congruent.
3. Step 4 Reason: Vertical angles are congruent.
4. Step 5 Reason: AAS (Angle - Angle - Side)
5. Step 6 Reason: Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
6. Step 7 Reason: Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
7. Step 8 Reason: In a triangle, sides opposite of congruent angles are congruent (or Corresponding Parts of Congruent Triangles are Congruent)