To determine which of the given conditions must be satisfied by congruent angles, we need to understand what congruent angles are.
Congruent angles are angles that have exactly the same angle measure. In other words, if two angles are congruent, they are equal in every way in terms of their size. Congruency of angles is strictly about the size of the angles and does not concern their orientation, position, or whether they share a common side or vertex.
Now, let's evaluate each of the given conditions:
A. They have the same angle measure.
- This statement is correct. Congruent angles, by definition, have the same angle measure. If two angles are 30 degrees each, they are congruent because they have the same measure.
B. The sum of their measures is 90°.
- This statement is incorrect concerning congruent angles. This condition describes complementary angles, not necessarily congruent angles. While it's possible that two congruent angles could sum to 90 degrees (if each angle is 45 degrees, for example), this is not a defining condition of congruency, and it wouldn't apply to angles with measures other than 45 degrees.
C. They share a vertex and a side.
- This statement is also incorrect concerning congruent angles. Sharing a common vertex and a side generally describes adjacent angles, not necessarily congruent angles. Congruent angles do not need to be adjacent; they could be anywhere in the plane or in space, as long as they have the same angle measure.
Given the analysis above, the correct condition that must be satisfied by congruent angles is:
A. They have the same angle measure.
