When θ = 15, 6 θ = 90. At this time, you can put the fifth stick, and θ should be larger according to the meaning of the question, so that the fifth stick can't be put down. Therefore, θ >: 15, you should have misread the answer, not 15 ≤ θ.

When 5 θ = 90 and θ = 18, the fourth stick can just be placed, so θ ≤ 18 meets the requirements. If θ is larger, the fourth stick cannot be placed.

Taken together, it is 15.

In your drawing, the circle Q and the ray AR should be tangent, that is to say, they can only be equal to 90. If it is greater than 90, as the angle you give in the figure, then there will be two intersections between the ray and the circle Q, which means that you can put two sticks at the same time with Q as the endpoint, which is not in line with the meaning of the question.