Suppose the angle drawn below has the measure of acute angle B in the isosceles
triangle that you're going to construct with straight edge and compass, and,
suppose the line segment drawn below the angle has the measure of the base of
that isosceles triangle.   

Bisect the angle, and construct the base of the triangle AC

Construct the perpendicular bisector of AC.
Construct a line perpendicular to AC at A:

Construct one half the given vertex angle with vertex at A,
and with the green line perpendicular to AC as its left side.

Extend the right side of the angle just constructed until it
intersects the perpendicular bisector of the base AC.  This
will be point B.

Draw in the third side of the triangle BC.  We know the construction
is correct because the three angles indicated by the red double arcs 
have equal measure:

 
Edwin