Question

given: \\(\overline{ab} \cong \overline{ae}; \overline{bc} \cong \overline{de}\\)
prove: \\(\angle acd \cong \angle adc\\)
complete the paragraph proof.
we are given \\(\overline{ab} \cong \overline{ae}\\) and \\(\overline{bc} \cong \overline{de}\\). this means abe is an isosceles triangle. base angles in an isosceles triangle are congruent based on the isosceles triangle theorem, so \\(\angle abe \cong \angle aeb\\). we can then determine \\(\triangle abc \cong \triangle aed\\) by \\(\square\\). because of cpctc, segment ac is congruent to segment \\(\square\\). triangle acd is an isosceles triangle based on the definition of isosceles triangle. therefore, based on the isosceles triangle theorem, \\(\angle acd \cong \angle adc\\).

Explanation:

Step1: Identify Triangle Congruence Criterion

We know \( \overline{AB} \cong \overline{AE} \), \( \overline{BC} \cong \overline{DE} \), and we found \( \angle ABE \cong \angle AEB \). So in \( \triangle ABC \) and \( \triangle AED \), we have two sides and the included angle congruent (SAS: \( AB = AE \), \( \angle ABC=\angle AEB \), \( BC = DE \)). So the congruence criterion is SAS.

Step2: Apply CPCTC

By CPCTC (Corresponding Parts of Congruent Triangles are Congruent), since \( \triangle ABC \cong \triangle AED \), the corresponding sides \( AC \) and \( AD \) are congruent. So segment \( AC \) is congruent to segment \( AD \).

Answer:

First blank: SAS (Side - Angle - Side)
Second blank: \( \overline{AD} \) (or \( AD \))