To determine which of the given conditions must be satisfied for angles to be congruent, let's carefully analyze each option:
Option OA: They have the same angle measure.
- For angles to be congruent, they must have the same angle measure. Congruence of angles specifically means that the angles are identical in measure. Therefore, two angles that have the same degree measure are congruent. For example, if one angle measures 45 degrees and another angle also measures 45 degrees, then these two angles are congruent.
Option OB: The sum of their measures is 90°.
- This condition pertains to complementary angles, not congruent angles. Complementary angles are two angles whose measures add up to 90 degrees. For instance, if one angle measures 30 degrees and another measures 60 degrees, their sum is 90 degrees, making them complementary but not necessarily congruent because their measures are different.
Option OC: They share a vertex and a side.
- This condition describes adjacent angles. Adjacent angles share a common vertex and a common side but are not necessarily congruent. They simply have a positional relationship without implying equality in measure. For example, two angles could share a vertex and a side but have different measures like 30 degrees and 60 degrees.
Based on the above analysis, the correct condition for the angles to be congruent is:
OA. They have the same angle measure.
Thus, the answer is OA.
