in a 13-sided polygon the sum of 5 angles is 1274 degree four of the eight angles remaining are equal and the other 4 18 degree less than each of the equal angles find the angles
Let's first find the sum of all the angles in a 13-sided polygon using the formula:
Sum of angles = (n-2) * 180 degrees, where n is the number of sides.
For a 13-sided polygon:
Sum of angles = (13-2) * 180 = 11 * 180 = 1980 degrees
Now we know that the sum of 5 angles is 1274 degrees. Let's call the four equal angles x, and the other four angles (which are 18 degrees less than x) y.
We can set up an equation:
5x + 4y = 1274
We also know that the sum of all 13 angles is 1980 degrees. So we can set up another equation:
5x + 4y + 4x + 4(y-18) = 1980
Simplifying this equation:
9x + 6y - 72 = 1980
9x + 6y = 2052
3x + 2y = 684
Now we have two equations with two variables:
5x + 4y = 1274
3x + 2y = 684
You can solve this system of equations using substitution or elimination. Here's one way to do it using elimination:
Multiply the second equation by 2 to get rid of the y variable:
5x + 4y = 1274
6x + 4y = 1368
Subtract the first equation from the second:
x = 94
Now we can substitute x back into one of the equations to find y:
5(94) + 4y = 1274
470 + 4y = 1274
4y = 804
y = 201
So the four equal angles are x = 94 degrees, and the other four angles are y = 201 degrees.
To check that these angles add up correctly, we can verify that:
5x + 4y = 5(94) + 4(201) = 1274
4x + 4(y-18) = 4(94) + 4(183) = 1368
5x + 4y + 4x + 4(y-18) = 9x + 8y - 72 = 1980
To find the angles in a 13-sided polygon, we can start by using the formula for the sum of the interior angles of a polygon:
Sum of interior angles = (n - 2) * 180,
where n is the number of sides of the polygon.
In this case, we have a 13-sided polygon, so the sum of the interior angles is
(13 - 2) * 180 = 11 * 180 = 1980 degrees.
We are given that the sum of 5 angles is 1274 degrees.
Let's denote the 5 angles as a1, a2, a3, a4, and a5.
a1 + a2 + a3 + a4 + a5 = 1274. [Equation 1]
We are also given that the remaining 8 angles can be divided into two groups:
- Four of the angles are equal, let's denote them as b1, b2, b3, and b4.
- The other four angles are each 18 degrees less than the equal angles, so they can be denoted as b1 - 18, b2 - 18, b3 - 18, and b4 - 18.
Let's find the sum of these remaining eight angles:
b1 + b2 + b3 + b4 + (b1 - 18) + (b2 - 18) + (b3 - 18) + (b4 - 18)
Simplifying this expression, we get:
2(b1 + b2 + b3 + b4) - 72
Since we know that the sum of interior angles is 1980 degrees, we can subtract the sum of the 5 angles we have already found:
1980 - 1274 = 706 degrees.
Now, we can set up an equation using the sum of the remaining eight angles:
2(b1 + b2 + b3 + b4) - 72 = 706. [Equation 2]
We have two equations now. We can solve them simultaneously to find the values of a1, a2, a3, a4, a5, b1, b2, b3, and b4.
To solve this problem, we need to break it down into smaller steps.
Step 1: Determine the sum of the angles in a 13-sided polygon.
The sum of the interior angles in any polygon can be found using the formula: (n-2) * 180, where n is the number of sides.
For a 13-sided polygon, the sum of the angles would be (13-2) * 180 = 11 * 180 = 1980 degrees.
Step 2: Find the sum of the five given angles.
We know that the sum of the five given angles is 1274 degrees.
Step 3: Find the sum of the remaining eight angles.
To find the sum of the remaining eight angles, subtract the sum of the five given angles from the total sum of angles in the polygon.
Remaining Angle Sum = Total Angle Sum - Sum of Given Angles
Remaining Angle Sum = 1980 - 1274
Remaining Angle Sum = 706 degrees.
Step 4: Find the measure of the equal angles.
Since there are four equal angles among the remaining eight angles, we can divide the remaining angle sum by four.
Equal Angle Measure = Remaining Angle Sum / 4
Equal Angle Measure = 706 / 4
Equal Angle Measure = 176.5 degrees.
Step 5: Find the measure of the other four angles.
The other four angles are each 18 degrees less than the equal angles. Therefore, we subtract 18 degrees from the measure of the equal angles to get the measure of the other four angles.
Other Angle Measure = Equal Angle Measure - 18
Other Angle Measure = 176.5 - 18
Other Angle Measure = 158.5 degrees.
So, the measure of the equal angles is 176.5 degrees, and the measure of the other four angles is 158.5 degrees.