1. The measure of an interior angle of a regular polygon is 20 more than thrice the measure of its adjacent exterior angle. Find the number of sides of the polygon and its total number of diagonals.
2. Find the sum and difference between the sum of the measures of the interior angles of a convex 23-gon and 14-gon.
The measure of each interior angle of a regular polygon is 20 more than three times the measure of each exterior angle. How many sides does the polygon have?
If the outer angle is 20 more than thrice the inner angle what is the measure of the inner angle
1. Wow, that's quite the math problem! It seems like this polygon is really playing hard to get with its angles. Let's break it down: let's call the measure of the interior angle "x" and the measure of the adjacent exterior angle "y". According to the question, we have the equation x = 3y + 20. Now, remember that the sum of the interior angles of a polygon is given by the formula (n-2) * 180, where "n" represents the number of sides. Since we're dealing with a regular polygon, all the interior angles are equal, so we can substitute x into the formula: x * n = (n-2) * 180. Now it's just a matter of substituting x = 3y + 20 and solving for n. Once we find n, we can use the formula for the number of diagonals (n * (n-3))/2 to find the total number of diagonals. Good luck!
2. Well, well, well, we have two polygons strutting their angles here! Let's sum up the sums of their measures, shall we? For the 23-gon, the sum of its interior angles is given by (23-2) * 180, and for the 14-gon, it's (14-2) * 180. Now, we just have to subtract these two sums to find the difference. But remember, math isn't always about differences, it's also about making friends, so let's calculate the sum and the difference separately. Go ahead, calculate those sums, then subtract one from the other, and you'll have the difference. And if you're feeling extra curious, you can also find the sum by adding both sums together. Happy trolling, I mean calculating!
1. To solve this problem, let's start by analyzing the relationship between the interior and exterior angles of a regular polygon.
In a regular polygon, all interior angles are equal and all exterior angles are equal. Let's denote the measure of an interior angle as x and the measure of its adjacent exterior angle as y.
Based on the given information, we can establish the following equation: x = 3y + 20.
Now, we need to find the value of x, which represents the measure of an interior angle. Since all the interior angles of a regular polygon are equal, we can create a relationship between the number of sides (n) and the interior angle measure. This relationship is given by the formula: x = (180 * (n - 2))/n.
Combining both equations, we have: (180 * (n - 2))/n = 3y + 20.
Simplifying this equation, we obtain: 180n - 360 = 3ny + 20n.
Now, we can solve for n. Rearranging the equation, we get: 180n - 20n = 3ny + 360.
Factoring out n on the right-hand side gives: n(180 - 20) = 3ny + 360.
Simplifying further, we have: 160n = 3ny + 360.
Dividing both sides by 3y + 360, we find: n = 160/(3y + 360).
Since the number of sides of a polygon must be a positive integer, we can start by substituting different values of y until we find an integer value for n.
2. To find the total number of diagonals in a polygon, we can use a formula that relates the number of sides (n) to the total number of diagonals (D). The formula is given by: D = (n * (n - 3))/2.
For the first part of the question, once you have found the value of n using the equation from step 1, you can substitute it into the formula for the total number of diagonals to find the answer.
2. To find the sum of the measures of the interior angles of a polygon, we can use the formula: S = (n - 2) * 180 degrees, where S represents the sum of the interior angles and n is the number of sides of the polygon.
For a 23-gon, the sum of the interior angles would be: S23 = (23 - 2) * 180 = 21 * 180 = 3780 degrees.
Similarly, for a 14-gon, the sum of the interior angles would be: S14 = (14 - 2) * 180 = 12 * 180 = 2160 degrees.
To find the difference between the sums of the interior angles, you can simply subtract the smaller value from the larger value.
Difference = S23 - S14 = 3780 - 2160 = 1620 degrees.
1. let the number of sides be n
each interior angle = 180(n-2)/n
exterior angle = 360/n
180(n-2)/n = 3(360/n) + 20
times n
180(n-2) = 1080 + 20n
180n - 360 = 1080 + 20n
160n = 1440
n = 9
We have a 9-gon, which is called a enneagon or nonagon
check:
interior angle = 180(7)/9 = 140
exterior angle = 360/9 = 40
is 3 times 40 + 20 = 140 ? , YES
2.
sum interior angles of 23-gon= 180(21) = 3780
sum of interior angles of 14-gon = ....
then take their sum and difference