A polygon is regular if all the sides are equal and all the interior

We have at least two inferences from the free info:
[*] Two sides of regular octagon O lie on lines l and m, respectively = lines l and m lie on two different sides of the octagon.
[*] Since the two lines intersect, they cannot lie on opposite sides (this would make the lines parallel and they would never intersect).

We can easily figure out the angle of intersection if we find out where one line lies, RELATIVE to the other.

The free info leaves us with three possibilities as shown in PICTURE 1. Pretend the blue line is line l and we have three possibilities for line m (red lines). The two lines can form either an angle given in A, B or C (the angles would be \(45^{\circ}\), \(90^{\circ}\), \(135^{\circ}\) resepectivly).

Statement 1
The lines can intersect in point A or B. The lines cannot intersect in point C, since this point is on the vertex of the octagon. INSUFFICIENT

Statement 2
Lines l and m can intersect in point A or C. The green lines in PICTURE 2 represent lines that bisect both its interior angle and the angle of intersections of lines l and m. INSUFFICIENT

Statements 1 and 2 together
The only point that satisfies both statements is point A. The angle here is \(45^{\circ}\). SUFFICIENT

The answer to the DS question is C.