I stumbled upon a problem in 6th grade geometry book.
The problem is as follows:
$AD\cong EC$ and $\angle ECD\cong \angle DAE$. We have to prove that $AB\cong BC$.
I stuck since we can not use angle sum in this problem.
Only things we can use, as I see, are Side-Angle-Side condition, Angle-Side-Angle condition or Side-Side-Side condition. Obviously we also can use properties of isosceles triangle which follows from the three conditions.
Does anyone see how to approach this problem?
Use AAS to prove the congruence of the bottom triangles. Then connect $\overline{AC}$ and use the properties of an isosceles triangle to prove the base angles $\angle EAC$ and $\angle ACD$ are congruent. In particular, the base angles of $\triangle ABC$ are congruent, and so it's isosceles too.