@the-blade-dancer Thank you for asking!

The reason choice ii.) doesn't work is because if you look at transversal line \(\overline{AC}\) which crosses the two parallel lines \(\overline{CD}\) and \(\overline{AB},\) we find that \(\angle \textcolor{blue}{PAB}\) is congruent to \(\angle \textcolor{blue}{PCD}.\) We don't know if \(\angle \textcolor{blue}{PCD}\) is the same as the angle \(\angle \textcolor{orange}{BCP},\) so that's why choice ii.) is false.

The symbol "\(\angle\)" means "angle," rather than triangle, so in choice iii.) it says that \(\angle CPB\) and \(\angle DPA\) are equal; it's not implying that the two triangles \(\bigtriangleup PDA\) and \(\bigtriangleup PCB\) are congruent.

I hope this helps! Please let us know if you have more questions!