"An angle has a measure three times that of its complement. What is the measure of the angle?"
Complementary angles add up to 90 degrees. If x is the complement and 3x is the angle, and you added them to have 4x, then you'd have 4x=90, right? That would then mean x=22.5 and 3x=67.5. They would then equal 90 degrees. What I am wondering is if I did this right, and if there can actually be an actual 67.5 degree angle. Anyone have any advice/solutions?
You have approached the problem correctly. The angle and its complement add up to 90 degrees, so if the angle is three times the measure of its complement, you can represent the angle as 3x and its complement as x.
To solve for x, you set up the equation:
3x + x = 90
Combining like terms, you get:
4x = 90
Dividing both sides by 4, you find:
x = 22.5
Since the complement is 22.5 degrees, the angle is three times that:
3(22.5) = 67.5 degrees.
Thus, according to the given problem, the angle measures 67.5 degrees.
In terms of whether an actual 67.5 degree angle can exist, in Euclidean geometry (traditional plane geometry), an angle can be any measurement between 0 degrees and 180 degrees. So, a 67.5 degree angle is within the valid range and can exist in this context.
Yes, you have correctly solved the problem. Let me explain in more detail:
Let's assume that x represents the measure of the complement angle. According to the problem, the given angle is three times that of its complement, so the measure of the angle would be 3x.
We know that complementary angles add up to 90 degrees. Therefore, we can write an equation: x + 3x = 90.
Combining like terms, we have 4x = 90.
To solve for x, we divide both sides of the equation by 4: x = 90/4 = 22.5.
Since we found that the complement angle measures 22.5 degrees, we can substitute this value back into the equation to find the measure of the angle: 3x = 3(22.5) = 67.5 degrees.
So, the measure of the given angle is indeed 67.5 degrees, which is a valid angle measurement.
Therefore, you have correctly solved the problem, and the measure of the angle is 67.5 degrees.
You did it right.
There is no reason an angle can not be measured in fractions of a degree.