Two angles are complementary. The sum of the measure of the first angle and half the second angle is 68 degrees. Find the measures of the angles:

What is the measure of the smaller angle?
What is the measure of the other angle?

To find the measures of the angles, let's assign variables to the angles. Let's call the first angle "x" and the second angle "y".

We know that two angles are complementary, which means their sum is 90 degrees. Therefore, we can write the first equation as:

x + y = 90 ---(Equation 1)

The problem also states that the sum of the measure of the first angle and half the second angle is 68 degrees. Mathematically, we can represent this as follows:

x + 0.5y = 68 ---(Equation 2)

Now, we have a system of two equations (Equation 1 and Equation 2) with two variables (x and y). We can solve this system of equations to find the values of x and y.

To solve the system, we can use the method of substitution or elimination. I will use the method of substitution here:

From Equation 1, we can express x in terms of y as follows:

x = 90 - y

Now, substitute this value of x into Equation 2:

90 - y + 0.5y = 68

Simplify the equation:

90 - 68 = 0.5y - y

22 = -0.5y

To isolate y, divide both sides of the equation by -0.5:

22 / -0.5 = y

y = -44

Now substitute this value of y back into Equation 1 to find the value of x:

x + (-44) = 90

x - 44 = 90

Add 44 to both sides:

x = 90 + 44

x = 134

So, the measure of the smaller angle (x) is 134 degrees, and the measure of the other angle (y) is -44 degrees. However, since we are dealing with angles, we cannot have negative angles. Therefore, we discard the negative solution.

The final answer is:

The measure of the smaller angle is 134 degrees.
The measure of the other angle is 90 - 134 = -44 degrees.

"Two angles are complementary."

If one angle is x, the other angle is 90-x.

"The sum of the measure of the first angle and half the second angle is 68 degrees."
x + (1/2)(90-x)=68

Solve for x, and hence find 90-x.

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