It is possible and to explain why consider the following question.
Is the angle formed by the red or by the green lines larger?
Trick question! The angles are identical.
You see, if we have our two points on the map and know the angle between the bearing from us to them, then we can use that information to find the circle that runs through our position and through those two points.
To define the circle, we need to find its center and its radius.
The easiest way to find the center is to find the point where the angle forms a right triangle with the two points. The center will be at the exact half of the hypotenuse (green line). Or one could do it twice from both sides and where the hypotenuses cross is the center.
Once we know the center, the radius is simply the distance between the center and either of the points.
To use that information to find our position we will need at least one more piece of information. One can use another circle with a third point and where the circles intersect is your location.
Of if you were on a long trail or other feature that went through the circle, that intercepted the circle, one could use logic to find your position. For example if you were south of the two positions in the image below, that would be enough to get a fix.
The easiest is when the angle is exactly 90 degrees, as the the center is halfway between the two points.