To construct an adyacent angle to another, and for it to be congruent we have to begin by tking the compass and drawing an arch we can call "\(\beta\) " in such a way that it intersects the given angle, let's call it \(\alpha\), in two points \(A\) and \(B\) which is any radius pretty much, let's now say that the two semilines whose origin lies on the vertice of angle \(\alpha\), are the semilines (l) and (r) respectively therefore point \(A\) lies on the semiline (l) and point \(B\) lies on the semiline (r).
With that said, we will take point \(B\) as a center, and radius \(AB\) and draw a semicircle we can call \(\delta\) on the line (r) in such way that it doesn't intersect the semiline (l).
Now, let's extend arch \(\beta\) and intersect it with arch \(\delta \) and it will define a new point \(C\) and this point will define another semiline (w) , that has it's origin on the vertex of angle \(\alpha \).
Then semilines (r) and (w) will define an angle congruent and adyacent to \(\alpha\).