The measures of all angles in a regular octagon are (4x + 12)°. Find the value of x. Show equations and all work that leads to your answer.
Hint:
In any regular polygon, the sum of all exterior angles is 180°
so .....
To find the value of x, we need to set up an equation using the given information.
In a regular octagon, all angles are equal. Let's call the measure of one angle in the octagon "A".
According to the given information, the measure of each angle in the octagon is (4x + 12)°. So, we have:
A = 4x + 12
Since an octagon has 8 angles, the sum of all the angles in the octagon can be calculated as follows:
Sum of all angles in the octagon = 8A
But we know that the sum of all angles in an octagon is equal to 180° less than a straight angle (a straight angle is 180°). Therefore:
8A = 180° - straight angle
Substituting the value of A from the first equation, we have:
8(4x + 12) = 180° - 180°
Simplifying:
32x + 96 = 0
Now, let's solve for x:
32x = -96
Dividing both sides by 32:
x = -3
So, the value of x is -3.
This means that (4x + 12)° will be (-3 * 4 + 12)°, which simplifies to 0°.
Therefore, each angle in the regular octagon is 0°.