From the image, find the value of alpha if line a is parallel to line b.

Let's analyze the angles formed when a transversal line c intersects two parallel lines a and b. In this case, we have two angles given: $$60^{\circ} + \alpha$$ and $$120^{\circ} - \alpha$$.

Since lines a and b are parallel, the interior angles on the same side of the transversal are supplementary, meaning they add up to $$180^{\circ}$$. Therefore, we can write the equation:

$$ (60^{\circ} + \alpha) + (120^{\circ} - \alpha) = 180^{\circ} $$

However, the equation above simplifies to $$180^{\circ} = 180^{\circ}$$, which does not help us find the value of $$\alpha$$. This suggests we are looking at corresponding angles. The angle supplementary to $$(120^{\circ} - \alpha)$$ is $$180^{\circ} - (120^{\circ} - \alpha) = 60^{\circ} + \alpha$$.

For the lines to be parallel, the given angles must satisfy the property of supplementary angles. Let's assume the angles are supplementary, meaning their sum is $$180^{\circ}$$:

$$(60^{\circ} + \alpha) + (120^{\circ} - \alpha) = 180^{\circ}$$

This simplifies to $$180^{\circ} = 180^{\circ}$$, which is always true regardless of the value of $$\alpha$$. This means we need to consider the case where the angles given are equal as corresponding angles when one of the angles is first converted to its corresponding angle.

Another possibility is that the angle $$60^{\circ} + \alpha$$ is equal to the vertically opposite angle of $$120^{\circ} - \alpha$$. In this case, the given angles are equal, then we have: $$60^{\circ} + \alpha = 120^{\circ} - \alpha$$ $$2\alpha = 120^{\circ} - 60^{\circ}$$ $$2\alpha = 60^{\circ}$$ $$\alpha = 30^{\circ}$$

Let's check if this value of $$\alpha$$ makes sense. If $$\alpha = 30^{\circ}$$, then the first angle is $$60^{\circ} + 30^{\circ} = 90^{\circ}$$, and the second angle is $$120^{\circ} - 30^{\circ} = 90^{\circ}$$. In this case, the lines a and b would be parallel.

Therefore, the value of $$\alpha$$ is $$30^{\circ}$$.

Ответ: 30