N the construction of copying an angle, how does step 5 ensure that the new angle will be congruent to the original angle?

When you make the first arc, at points x and y in this article, you are actually marking the lengths of two sides of a triangle. When you draw the same arc on your new angle, you are transferring the same lengths. Then, the third side of this triangle is the line segment from x to y (for this construction, we never really draw that line segment XY). When you set your compass to the XY distance and then draw that arc on the new diagram, you are marking the third side of that virtual triangle. When two triangles have all three sides the same lengths, then all parts of that triangle are congruent as well. (This is a proof that you can work out for yourself.) Therefore, this construction focuses only on a single congruent angle, but in fact you will create a congruent triangle (if you connect points XY and points FG).