Question: I have been looking for the answer to this question but I haven’t found anything related.
I will use the following image as an example.

![Image](https://i.stack.imgur.com/ubPBJ.jpg)

As you can see, there is a circle divided in different parts of the same size and their corresponding angle is indicated.
So what do I mean with “getting the opposite angle”?
Based on the image, let’s say I want to get the opposite angle of 60.
While searching I have found that there’s one way to get the opposite angle, but it’s not exactly what I need. Those answers said that to get it I should add 180 to 60, but the problem is that it gives me 240, and that’s not what I need.
I should get 120 as the value, the opposite horizontal angle.
Is there any way to get this?

Answer: 
It is the supplementary angle with a common sine value of the given angle.

$$ \sin x =\sin \alpha$$

$$ x\_1= \alpha, x\_2= 180^{\circ} -\alpha$$

It is not on the opposite side in the diagram you have given. It is mirrored with respect to y-axis.
When you look at a pair of intersecting lines, you can see that four angles are formed around the point of intersection. The two angles that lie on the same line form an adjacent pair (summing to $180^{o} $), while the two that lie on different sides of the same line (facing each other) form a pair of opposite angles. With this in mind, had you named the points on the circle and its center, you could have been able to find it. The angles measure the same , then.

I said only what I know as a teen. Please let me know if this is correct.