The interior angle measures of a hexagon are in the ratio of 2:3:3:3:3:4. What is the degree measure of the largest angle?
sum of interior angles is 4*180 = 720
you have 18 parts.
720/18 = 40
4*40 = 160
To find the degree measure of the largest angle in the hexagon, we first need to find the sum of all the interior angles.
The sum of the interior angles of a hexagon can be found using the formula: (n-2) * 180 degrees, where n is the number of sides of the polygon. In this case, n = 6 (since it is a hexagon), so we have:
(6-2) * 180 = 4 * 180 = 720 degrees
Next, we need to find the value of each ratio. Let's assign variables to represent these values:
Let x be the measure of the first ratio (2x)
Let y be the measure of the second ratio (3y)
Let z be the measure of the third ratio (3z)
Let a be the measure of the fourth ratio (3a)
Let b be the measure of the fifth ratio (3b)
Let c be the measure of the sixth ratio (4c)
The sum of all these angles should be equal to 720 degrees:
2x + 3y + 3z + 3a + 3b + 4c = 720
We are given that the ratios are in the form of 2:3:3:3:3:4. Using this information, we can set up the following equations:
2x = 2y = 2z = 2a = 2b = 2c
3x = 3y = 3z = 3a = 3b
4x = 4c
From these equations, we can solve for the variables. However, notice that the value of x does not affect the sum of the angles since it is constant across all the angles. Therefore, we can ignore it and focus on the other variables.
Solving the equations:
3x = 3y = 3z = 3a = 3b (equation 1)
4x = 4c (equation 2)
From equation 1, we can solve for y, z, a, and b in terms of x:
3y = 3x
y = x
3z = 3x
z = x
3a = 3x
a = x
3b = 3x
b = x
Now, substitute these values into equation 2:
4x = 4c
Dividing by 4 on both sides,
x = c
Now, we can substitute these values back into equation 1:
3y = 3x
3y = 3(c)
y = c
From this, we can see that all the variables are equal to c, which means the ratio is effectively 2x:3x:3x:3x:3x:4x, where x = c.
Substituting these values back into the equation for the sum of the angles:
2x + 3y + 3z + 3a + 3b + 4c = 720
2x + 3x + 3x + 3x + 3x + 4x = 720
18x = 720
Dividing both sides by 18:
x = 40
Since the largest angle is represented by 4c, we can find its degree measure by multiplying x (or c) by 4:
4c = 4 * 40 = 160 degrees
Therefore, the degree measure of the largest angle in the hexagon is 160 degrees.
To find the degree measure of the largest angle, we need to determine the sum of all the interior angle measures of the hexagon.
A hexagon has six interior angles. Let's assign variables to the angle measures:
Let the angles be 2x, 3x, 3x, 3x, 3x, and 4x, respectively.
The sum of all the interior angles of a hexagon is given by the formula:
Sum = (6 - 2) * 180°, since each interior angle of a hexagon is supplementary to the adjacent angle.
Simplifying the equation, we have:
Sum = 4 * 180° = 720°
Now, setting up an equation for the sum of the angle measures:
2x + 3x + 3x + 3x + 3x + 4x = 720°
Adding like terms:
18x = 720°
Dividing both sides by 18:
x = 40°
Now, we can find the degree measure of the largest angle, which is 4 times x:
4 * 40° = 160°
Therefore, the degree measure of the largest angle in the hexagon is 160°.