If the measure of each interior angle of $a$ regular polygon is 150°, then the number of its diagonals will be ________.

Step 1 (Find Exterior Angle): In a regular polygon, the sum of an interior angle and its corresponding exterior angle is $180^{\circ }$. We are given that each interior angle is $150^{\circ }$:

Step 2 (Determine Number of Sides n): The sum of all exterior angles in any convex polygon is always $360^{\circ }$. For a regular polygon with $n$ sides:

So the polygon is a regular dodecagon (12 sides).

Step 3 (Calculate Number of Diagonals): The formula for the number of diagonals in a polygon of $n$ sides is:

Substituting $n=12$:

Step 4 (Conclusion): The regular polygon has exactly 54 diagonals.