How many sides does this polygon have ?

The sum of exterior angles is $360^\circ$

The smallest exterior angle is $180^\circ-153^\circ=27^\circ$

Start adding $27+29+..$ (I used a spreadsheet to make it easy) and it takes 10 numbers to get to $360^\circ$ , so the answer is $\boxed{10}$ .

Sum of interior angles of this polygon = 153+151+149+…(153–2(n−1)) = (n–2)∗180

If there are n sides, there are n interior angles.

The second largest angle will be 153 – 2*1. (Since the angles are odd.)

The third largest will be 153 – 2*2.

The smallest will be 153 – 2*(n-1).

This is an arithmetic progression.

Sum of all terms = n (First-term+Last-term)/2 = n (153 +153 – 2(n−1))/2

Solving this you get, n = 10