the sum of the measures of an obtuse angle and acute angle is 12 degrees.the complement of the acute angle and one-half the suplement of the obtuse angle is 75 degree.find the measures of the two angles.
typo.
Since any obtuse angle is greater than 90 degrees, how can the sum be 12?
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To find the measures of the two angles, we can set up a system of equations based on the given information.
Let's assume:
- The measure of the obtuse angle is x degrees.
- The measure of the acute angle is y degrees.
According to the problem, the sum of the measures of the obtuse angle and acute angle is 12 degrees. So, we can write the equation:
x + y = 12 ---(Equation 1)
The complement of an angle is the difference between 90 degrees and the angle. Therefore, the complement of the acute angle is (90 - y) degrees.
Half the supplement of an angle is half of the difference between 180 degrees and the angle. So, half the supplement of the obtuse angle is (1/2)(180 - x) degrees.
According to the problem, the complement of the acute angle and half the supplement of the obtuse angle is 75 degrees. So, we can write the equation:
(90 - y) + (1/2)(180 - x) = 75 ---(Equation 2)
We now have a system of two equations (Equation 1 and Equation 2) with two variables (x and y). We can solve this system of equations simultaneously to find the values of x and y.
Let's solve the system of equations:
Equation 1: x + y = 12
Equation 2: (90 - y) + (1/2)(180 - x) = 75
First, let's simplify Equation 2:
90 - y + (1/2)(180 - x) = 75
90 - y + 90 - (1/2)x = 75
180 - y - (1/2)x = 75
- y - (1/2)x = 75 - 180
- y - (1/2)x = -105
Now, let's multiply the entire equation by -2 to eliminate the fraction:
2(y) + (1)x = 210
We now have a system of two equations:
x + y = 12
2y + x = 210
Using any preferred method (substitution, elimination, matrices), you can solve this system of equations to find the values of x and y.
After solving, you will find that:
x = 108 degrees
y = -96 degrees
However, an angle cannot have a negative measure, so the value of y is not possible in this case.
Hence, only one angle can be determined, which is the measure of the obtuse angle:
The measure of the obtuse angle is x = 108 degrees.