Question

given the figure shown to the right, write a two - column proof to prove \\( \angle cae\cong\angle cea \\).

Explanation:

Step1: Prove $\triangle ABC\cong\triangle EDC$

• $\angle BAC\cong\angle DEC$ (Given, from the figure, the non - marked angles in the two triangles adjacent to $BC$ and $CD$ are equal)
• $\angle ABC\cong\angle EDC$ (Given)
• $BC\cong CD$ (Given)
• By the AAS (Angle - Angle - Side) congruence criterion, $\triangle ABC\cong\triangle EDC$.

Step2: Get $AC\cong EC$

• Since $\triangle ABC\cong\triangle EDC$, by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), $AC = EC$.

Step3: Prove $\angle CAE\cong\angle CEA$

• In $\triangle ACE$, if $AC\cong EC$, then by the definition of an isosceles triangle (a triangle with two equal sides has two equal angles opposite those sides), $\angle CAE\cong\angle CEA$.

Answer:

Statement	Reason
$\triangle ABC\cong\triangle EDC$	AAS
$AC\cong EC$	CPCTC
$\angle CAE\cong\angle CEA$	Definition of isosceles triangle