It seems you're asking which theorem applies to a specific scenario or question you may have in mind, but you haven't provided the context or details of the scenario. Here is a brief overview of each theorem you mentioned:
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Vertical Angles Theorem: States that vertical angles (the angles opposite each other when two lines intersect) are equal.
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Triangle Sum Theorem: States that the sum of the interior angles of a triangle is always 180 degrees.
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Converse of Alternate Interior Angles Theorem: States that if two lines are cut by a transversal and the alternate interior angles are equal, then the lines are parallel.
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Perpendicular Bisector Theorem: States that any point on the perpendicular bisector of a segment is equidistant from the endpoints of that segment.
If you provide a specific scenario or set of angles, I can help you determine which theorem applies to that case!