Question

The person who can answer this question gets to suggest what I should change my profile picture to. If you show your work, I will change it to whatever you say.

Answer 1

The sum of exterior angles of ANY regular polygon is 360.
For example, the exterior angle of a square is 90 degrees time 4 vertices = 360 degrees.
The exterior angles of an equilateral triangle is 180-60=120, times 3 vertices = 360.
...and so on.

So we can use this fact to calculate the exterior angle of a REGULAR decagon, which is therefore 360/10=36 degrees.

According to the diagram, this angle (36 degrees) is the complement of x, so x=90-36=54 degrees.

Next, the interior angle is just the supplement of the exterior angle, namely 
y=180-36=144 degrees.

BTW, if you would grant my wish, please put a Micky Mouse as your profile picture, and that, of course.....until you change it next time!   :)