It looks like you've uploaded an image with a geometry problem. I'll take a closer look and provide a solution or explanation for the question.

Let me break down the details of the problem:

1. Part (1) asks to prove that $BD \perp BC$, given that $CD \parallel EF$, $A$ and $B$ lie on $CD$ and $EF$ respectively, $BC$ bisects $\angle ABE$, and $BD$ bisects $\angle ABF$.

2. Part (2) is about finding the angle $\angle APD$, with point $G$ lying on $BF$ and $\angle BAG = 50^\circ$. The angle bisector of $\angle DAG$ intersects $BD$ at point $P$.

3. Part (3) involves finding $\angle PAQ$, given that $AN \perp EF$, $AQ \parallel BC$, $EF$ intersects $BC$ at point $Q$, and the angle bisector of $\angle BAN$ intersects $BF$ at point $P$.

I'll work through these steps methodically to give you a detailed solution.

Would you like to start with a specific part, or would you like me to proceed step by step with all parts? Let me know!