Question

statement reason $\angle ace \cong \angle acf$ $\angle ecb \cong \angle fcd$ given $\overline{bc} \cong \overline{dc}$ $\angle acb \cong \angle acd$ substitution $\overline{ac} \cong \overline{ac}$ reflexive property $\triangle abc \cong \triangle adc$ sas $\overline{ab} \cong \overline{ad}$ corresponding parts of congruent triangles are congruent (cpctc) the proof is complete. you may now press submit. you can still change reasons.

Explanation:

Step1: Analyze the given congruent angles

We know that \(\angle ACE\cong\angle ACF\) and \(\angle ECB\cong\angle FCD\). By the angle - addition postulate, \(\angle ACB=\angle ACE+\angle ECB\) and \(\angle ACD = \angle ACF+\angle FCD\). Since \(\angle ACE\cong\angle ACF\) and \(\angle ECB\cong\angle FCD\), we can substitute these congruent angles into the angle - addition expressions. So, \(\angle ACB\cong\angle ACD\) (by substitution of congruent angles).

Step2: Consider the side - side - angle (SAS) criterion

We are given that \(\overline{BC}\cong\overline{DC}\), we just proved \(\angle ACB\cong\angle ACD\), and \(\overline{AC}\cong\overline{AC}\) (by the reflexive property of congruence, which states that any segment is congruent to itself).
For two triangles \(\triangle ABC\) and \(\triangle ADC\), in the SAS (side - angle - side) 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. Here, in \(\triangle ABC\) and \(\triangle ADC\), \(\overline{BC}\cong\overline{DC}\), \(\angle ACB\cong\angle ACD\), and \(\overline{AC}\cong\overline{AC}\). So, \(\triangle ABC\cong\triangle ADC\) (by SAS).

Step3: Use the CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle ABC\cong\triangle ADC\), when two triangles are congruent, their corresponding parts (sides and angles) are congruent. For the sides \(\overline{AB}\) and \(\overline{AD}\), which are corresponding sides of \(\triangle ABC\) and \(\triangle ADC\) respectively, we have \(\overline{AB}\cong\overline{AD}\) (by CPCTC).

Answer:

The proof is correct as it uses substitution to get \(\angle ACB\cong\angle ACD\), the reflexive property for \(\overline{AC}\cong\overline{AC}\), the SAS criterion to prove \(\triangle ABC\cong\triangle ADC\), and then CPCTC to show \(\overline{AB}\cong\overline{AD}\).