The angle β and the line segments a and B are known. How many triangles can be made by using a ruler to make the angle B = angle β, BC = a, AC = B? Angle β is an acute angle Line segment a is longer than B Better have a picture!

The angle β and the line segments a and B are known. How many triangles can be made by using a ruler to make the angle B = angle β, BC = a, AC = B? Angle β is an acute angle Line segment a is longer than B Better have a picture!

Start with angle MBN (for a clearer look, it's better to draw MB horizontally and Nb upward)
Then cut BC = a on Nb. Draw an arc with C as the center and B as the radius. When B = asinb, the arc is tangent to MB (that is, there is only one intersection point)
Such a triangle can and can only be one
When B is less than asinb, the arc is separated from MB (i.e. there is no intersection point)
Such a triangle can not be made, that is, it can be made 0
When B is greater than asinb, the arc intersects MB (i.e. there are two intersections)
Such a triangle can be made into two
You should be able to understand my statement without pictures. I hope you will be satisfied