Question

Question 10 of the picture

Answer 1

This question is pretty hard, but not impossible.
The key to solving this is to spot triangles.

Since we want to prove that ∠XBP and ∠YBQ are equal, we need to know their relationship towards one another.
If we look at Δs PQB and XYB, we can note down their relationship.

Thus, the way to begin is to prove either similar or congruent triangles.
In Δs PQB and XYB,
∠QPB = ∠YXB (angles in the alternate segment standing on the same arc, PX, are equal)
Similarly, ∠XYB = ∠PQB ("  ^  ") note: this means see reasoning above

Since we proved two angles are equal, by definition, the third angle must be equal.
Thus, ∠PBQ = ∠PBY (by definition, they are equal)
Since the two angles share a common angle between them, namely the angle marked on the diagram, then this means ∠XBP must be equal to ∠YBQ, as required.