Please solve the attached proof.
Hello jwpaul
Welcome to Math Help Forum!
Sorry, but this question doesn't make sense.
If $\displaystyle \angle1 \cong \angle 2$, then $\displaystyle \triangle ETS$ is isosceles, with $\displaystyle ET = TS$.
Similarly $\displaystyle \triangle PAT$ is isosceles, with $\displaystyle PT = TA$.
Then $\displaystyle PS = PT+TS = AT + TE = AE$.
Grandad