A regular polygon has an exterior angel measure of (x+3) Degrees and an interior angle measure of (13x-33) degrees
Find the measure of each angle
How mant sides does this polygon have?
Look at a pair of interior and exterior angles.
Don't they form a straight line, so
13x-33 + x+3 = 180
14x = 210
x = 15
each exterior angle is 18°
and each interior is 162°
the sum of all exterior angles is always 360
so each angle is 360/18 or 20 angles
20 angles ----> 20 sides
To find the measure of each angle of a regular polygon, we can use the formula:
Interior angle = (n-2) * 180 / n
where n is the number of sides of the polygon.
Given that the interior angle measure is (13x - 33) degrees, we can set up the equation:
(13x - 33) = ((n - 2) * 180) / n
Now, let's solve this equation to find the value of x, and then we can find the measure of each angle and the number of sides.
1. Start by distributing n to the terms inside the parentheses:
13x - 33 = (180n - 360) / n
2. Multiply through by n to eliminate the fraction:
n(13x - 33) = 180n - 360
13nx - 33n = 180n - 360
3. Move all the terms involving x to one side of the equation:
13nx - 180n = 33n - 360
4. Factor out x:
x(13n - 180) = 33n - 360
5. Divide both sides of the equation by (13n - 180) to isolate x:
x = (33n - 360) / (13n - 180)
Now we need to find the value of n, the number of sides:
The exterior angle of a regular polygon is the supplementary angle to the interior angle, which means it has a measure of 180 - (13x - 33) degrees = 213 - 13x degrees.
Given that the exterior angle measure is (x + 3) degrees, we can set up the equation:
(x + 3) = 213 - 13x
Now solve this equation for x.
1. Collect like terms:
14x + 3 = 213
2. Subtract 3 from both sides:
14x = 210
3. Divide both sides by 14:
x = 15
Now substitute the value of x back into the equation to find the number of sides:
n = (33n - 360) / (13n - 180)
n = (33n - 360) / (13n - 180)
n = (33n - 360) / (13n - 180)
n = (33n - 360) / (13n - 180)
n = (33 * 15 - 360) / (13 * 15 - 180)
n = (495 - 360) / (195 - 180)
n = 135 / 15
n = 9
Therefore, the measure of each angle in the regular polygon is:
Interior angle = (13x - 33) = (13 * 15 - 33) = 162 degrees
Exterior angle = (x + 3) = (15 + 3) = 18 degrees
The polygon has 9 sides.