Find intersection of two lines given subtended angle

I’m tinkering with a bit of graphics software.

I want to be able to nudge the value of a single angle in a polygon, then redraw the new polygon.

The intersection point of the two lines creating the angle will change, and their lengths will change, but I can’t work out how to calculate the new coordinates.

All other intersections in the polygon will be unchanged as to position. (Obviously, the angles of the adjacent corners shall have to be modified as well as the corner that I explicitly “nudge.”)

In summary:

Find the coordinates of the point of intersection of two lines, given a single coordinate pair for each line and the measure of the angle of intersection.

Geometrically I think there will be more than one possible coordinate pair for any angle other than $180$ degrees. If I can choose the coordinate pair nearest to the previous coordinate pair, I think that would give the result I seek. But, I’d be happy just to know what general approach to take.

This problem is not the same as finding intersection for lines where both slopes are known.

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The point of intersection cannot be uniquely determined. This is because of the “angle in the same segment” theorem. See below.