Angle between two rays given, to find a curve with constant sum of intersection angles

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Two given lines $AB, AC$ including $\angle A= 60^{\circ}$ between them are intersected by a variable smooth continuous curve (red) so that angle sum $ \beta+\gamma = 150^{\circ} $ is always constant.

How do we find such a curve, assuming it exists?

If instead the angle sum is $ \beta+\gamma = 120^{\circ} $ then we have all moving transversal straight lines. Thanks in advance.

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