Question

given the figure shown to the right, write a two - column proof to prove \\( \angle cae\cong\angle cea \\).
complete the proof below.
statement
\\( \angle bac\cong\angle dec,\angle abc\cong\angle edc,\overline{bc}\cong\overline{cd} \\)
\\( \triangle abc\cong\triangle edc \\)
\\( \overline{ac}\cong\overline{ec} \\)

Explanation:

Step1: Use AAS (Angle - Angle - Side) congruence

Since \(\angle BAC\cong\angle DEC\), \(\angle ABC\cong\angle EDC\), and \(BC\cong CD\), by the AAS (Angle - Angle - Side) congruence criterion, \(\triangle ABC\cong\triangle EDC\).

Step2: Use CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

If \(\triangle ABC\cong\triangle EDC\), then their corresponding parts are congruent. So, \(AC\cong EC\) (CPCTC).

Step3: Use the property of isosceles triangle

In \(\triangle CAE\), since \(AC\cong EC\), by the property that in an isosceles triangle, the angles opposite to the equal sides are equal, \(\angle CAE\cong\angle CEA\).

Answer:

The reason for \(AC\cong EC\) is CPCTC.