Hi!
The triangle pictured is an isosceles triangle. Isosceles triangles all have two sides of the same length and a third side, with the angles opposite the sides of the same lengths called the base angles. The base angles have the same measure, and it's only the third angle that has a different one.
In this case, the base angles are X and Z, because they are opposite the sides that are the same length in the triangle. Therefore, angle X has the same measure of angle Z.
A triangle's angles always add up to 180 degrees. Therefore, you know that m∠Y + m∠X + m∠Z = 180. And since you know m∠Y is 52 degrees, then you can say:
52 + m∠X + m∠Z = 180
And since m∠X is equal to m∠Z:
52 + m∠X + m∠X = 180
m∠X * 2 = 180 - 52
m∠X * 2 = 128
Divide everything by 2:
m∠X = 64 degrees