It's impossible to trisect an arbitrary angle with compass and straightedge alone.
But origami can do it! This uses Abe's method (Huzita-Hatori Axiom 6).
0 Show the angle to trisect
1 Mark bottom edge and create horizontal fold line
2 Create midpoint fold line
3 Apply Axiom 6: fold two points onto two lines simultaneously
4 The fold crease creates the trisection lines
5 Verify: three equal angles!