Question

step
statement
reason
1
\\( \angle b c f \cong \angle d c f \\)
\\( \overline{b c} \cong \overline{d c} \\)
\\( c \\) is the midpoint of \\( \overline{a e} \\)
\\( \angle a c f \\) is a right angle
\\( \angle e c f \\) is a right angle
given
2
\\( \angle a c b \\) and \\( \angle b c f \\) are complementary
if two angles form a right angle, then they are complementary
3
\\( \angle e c d \\) and \\( \angle d c f \\) are complementary
if two angles form a right angle, then they are complementary
4
\\( \angle a c b \cong \angle e c d \\)
if two angles are complements of the same angle (or congruent angles),
then they are congruent
5
6
\\( \triangle a c b \cong \triangle e c d \\)
sas

Explanation:

Step1: Identify the congruent sides

Since \(C\) is the mid - point of \(\overline{AE}\), we have \(\overline{AC}\cong\overline{EC}\) (by the definition of a mid - point).

Step2: Recall the given congruent angles and sides

We are given \(\angle BCF\cong\angle DCF\), \(\overline{BC}\cong\overline{DC}\), \(\angle ACB\cong\angle ECD\) (from step 4 of the given proof), and now \(\overline{AC}\cong\overline{EC}\) (from step 1).

Step3: Apply the SAS (Side - Angle - Side) congruence criterion

For \(\triangle ACB\) and \(\triangle ECD\), we have:

• Side: \(\overline{AC}\cong\overline{EC}\)
• Angle: \(\angle ACB\cong\angle ECD\)
• Side: \(\overline{BC}\cong\overline{DC}\)

By the SAS congruence criterion, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Answer:

\(\triangle ACB\cong\triangle ECD\) (by the SAS congruence criterion)