Final answer:
The interior angle x° of a regular polygon is found to be 144 degrees by solving a supplementary angle equation. Subsequently, the number of sides of the polygon is calculated to be 10, making it a decagon.
Step-by-step explanation:
The problem involves finding the magnitude of the interior angle (x°) of a regular polygon and then determining the number of sides of the polygon based on the relationship between interior and exterior angles.
Since the problem states that each exterior angle is (x-36)/3, and we know that the interior angle (x°) and its corresponding exterior angle are supplementary, meaning they add up to 180 degrees, we can set up the following equation:
x + (x-36)/3 = 180
Multiplying through by 3 to eliminate the fraction gives us:
3x + x - 36 = 540
Combining like terms leads to:
4x - 36 = 540
Adding 36 to both sides we get:
4x = 576
Dividing both sides by 4:
x = 144°
That is the magnitude of the interior angle. Now, to find the number of sides (n) of the polygon, we use the formula for finding the sum of interior angles of a polygon:
(n - 2) × 180°
Then we set that equal to the product of the number of sides and the magnitude of each interior angle:
(n - 2) × 180° = n × 144°
Dividing through by 36 to simplify:
(n - 2) × 5 = n × 4
Expanding the equation, we get:
5n - 10 = 4n
Subtracting 4n from both sides we find:
n = 10
The polygon has 10 sides, which means it is a decagon.