While practicing mathematics Olympiad questions , I got the below given question . Though the solution is given , I am not able to bypass certain steps... Attachment 46815
Can anyone please explain me why angle KPA and angle KQA are each equal to 90-angle A ? I am not able to get this step .
I have got the following..

Attachment 46817

]It is evident from the figure that angle KPA = angle CPA - angle XPK = 90 - angle XPA. So does this mean that angle XPK = angle A ? If yes , plz explain how ?
Awaiting for the get through...

To show that angle KPA = angle KQA = 90 - angle PAQ, note that PK is perpendicular to AQ and intersects AQ at, say, R. Similarly, QK is perpendicular to AP and intersects AP at, say, S. Triangles RPA and SQA are right triangles.