Two angles can have exactly five points in common.
Step 1: An angle is made of its arms (two rays) and the region between them. It has infinitely many points.
Step 2: Draw two angles that overlap. For example:
Write the first angle as \( \angle AOB \).
Write the second angle as \( \angle COD \).
Step 3: Make them overlap partly (same vertex or overlapping interiors). The part that overlaps is common to both angles.
Step 4: In that common part, mark five distinct points:
\( P, \; Q, \; R, \; S, \; T \)
Each of these points lies in \( \angle AOB \) and also in \( \angle COD \).
Conclusion: Since we can find five distinct common points in the overlap, the statement “Two angles can have exactly five points in common” is true.