In a regular polygon each exterior angle is 90°less than each interior angle. Calculate the number of of sides of the polygon hence give its name
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To begin, let's set up an equation based on the given information.
Let's assume that the interior angle of the regular polygon is "x" degrees.
According to the information given, each exterior angle is 90 degrees less than each interior angle. Thus, each exterior angle would be (x - 90) degrees.
Now, we can use the sum of the exterior angles in a polygon to find the number of sides. The sum of the exterior angles in any polygon is always equal to 360 degrees.
In a regular polygon, where all the exterior angles are equal, we can use the formula:
Sum of Exterior Angles = Number of Sides × Each Exterior Angle
Therefore, we have:
360 = Number of Sides × (x - 90)
Now, we can solve this equation for the number of sides (Number of Sides):
360 = Number of Sides × x - Number of Sides × 90
Rearranging the terms:
360 = x × Number of Sides - 90 × Number of Sides
Combining like terms:
360 = (x - 90) × Number of Sides
Now, we know that (x - 90) equals each exterior angle.
So, we can substitute the value of (x - 90) in terms of each exterior angle into the equation:
360 = Each Exterior Angle × Number of Sides
Since we already know that each exterior angle is given as (x - 90) degrees, we can substitute this value into the equation:
360 = (x - 90) × Number of Sides
Now, we can solve for the number of sides (Number of Sides) by dividing both sides of the equation by (x - 90):
(360 ÷ (x - 90)) = Number of Sides
To find the value of "x" and determine the number of sides, we need additional information or constraints about the polygon. Without a specific constraint, we cannot definitively determine the number of sides or the name of the polygon.
x + x-90 = 180
x = 135
This is true for an n-gon if
(n-2)*180 = n*135
Now finish it off