If [tex] \overline{AC} \simeq \overline{AD} [/tex], then [tex] ACD [/tex] is isosceles, and its base angles are congruent. So, angles [tex] ACD [/tex] and [tex] ADC [/tex] are congruent.
If [tex] \overline{BE} \simeq \overline{BC} [/tex], then [tex] BCE [/tex] is isosceles, and its base angles are congruent. So, angles [tex] BCE [/tex] and [tex] BEC [/tex] are congruent.
In the third case, [tex] ABE [/tex] is isosceles, and the side adjacent to the congruent angles are congruent, so [tex] \overline{AB} \simeq \overline{AE} [/tex]
Similarly, in this case, [tex] CED [/tex] is isosceles, and the side adjacent to the congruent angles are congruent, so [tex] \overline{CD} \simeq \overline{CE} [/tex]
