adjacent angle
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Yes (supposing that by side you mean ray...angles don't have sides because they are 2D, not 3D.) If you have a point and you draw four random rays from that point, that only meat at that point, then you would have created four angles. The two angles on opposite sides of the point will share a vertex, but not a RAY.
Adjecent Angles
Adjacent angles share a vertex and one ray, while any two angles that don't share both those things are nonadjacent.
Two angles that share a common side and add up to 180 degrees are called right angles and are 90 degrees.
opposite angles