(I) In\[\vartriangle APB\text{ }and\text{ }\vartriangle CPD\], we have
\[\angle APB\text{ }=\angle CPD\] [Vertically inverse angles]
\[\angle ABP\text{ }=\angle CDP\] [Alternate points as, AB||DC]
Consequently, \[\vartriangle APB\text{ }\sim\text{ }\vartriangle CPD\]by \[AA\] similarity criterion.
(ii) As \[\vartriangle APB\text{ }\sim\text{ }\vartriangle CPD\]
Since the comparing sides of comparative triangles are relative, we have
\[PA/PC\text{ }=\text{ }PB/PD\]
Consequently,
\[PA\text{ }x\text{ }PD\text{ }=\text{ }PB\text{ }x\text{ }PC\]