Question

part 5 of 5
given the figure shown to the right, write
a two - column proof to prove \\( \angle cae \cong \angle cea \\).
complete the proof below.
\\( \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} \\)
\\( \triangle ace \\) is isosceles.
\\( \angle cae \cong \angle cea \\)

Explanation:

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

By the AAS (Angle - Angle - Side) congruence criterion (since $\angle BAC\cong\angle DEC$, $\angle ABC\cong\angle EDC$, and $BC\cong CD$).

Step2: Justify $\overline{AC}\cong\overline{EC}$

Corresponding parts of congruent triangles are congruent (CPCTC).

Step3: Justify $\triangle ACE$ is isosceles

A triangle with two congruent sides ($\overline{AC}\cong\overline{EC}$) is isosceles.

Step4: Justify $\angle CAE\cong\angle CEA$

By the Isosceles Triangle Theorem (in an isosceles triangle, the angles opposite the congruent sides are congruent).

Answer:

Isosceles Triangle Theorem