Not quite sure how to reply to this post.
You have gotten the correct answer.
The fact that I would have done it differently does not change anything.
For this question i thought that angle 1 was 109 degrees. I drew lines from the center to angle A and B which i made 90 degrees. Then i split 38 in half to get 19 to make 2 90 degree angels. Then i solved for the remaining angle (AOP) which gave me 71. Then with 71+90+?=180 (sum of a triangle) which gave me 54.5. Which was also the same for the other side as they are equal. so I times 54.5 by 2 to get 109. I just dont know if i did it right and what i would say is my proof for finding that? Or if there is a way faster way of solving that?
You can also arrive at the same answer by drawing a line straight from A to B.
this makes the triangle ABP.AP and BP are equal so angles ABP and BAP are equal.
angle in a triangle add upp to 180 so 180-38=142. so angles ABP and BAP are 71.
Using the alternate segment theorem the angle the tangent makes with the left side of the triangle is equal to angle 1 so 180-71=109 is angle 1.
Also it is good that you look for alternate ways to solve problems.