Questions: What is the measure of an angle if it is one-fourth of its supplement? a. 36 degrees b. 45 degrees c. 54 degrees d. 144 degrees The measures of two complementary angles are in the ratio 4:5. What are the measures of the two angles? a. 30 degrees and 60 degrees b. 36 degrees and 54 degrees c. 40 degrees and 50 degrees d. 45 degrees and 45 degrees

Solution Steps

To solve these problems, we need to use the properties of supplementary and complementary angles.

1. For the first question, let the measure of the angle be \( x \). Its supplement will be \( 180 - x \). According to the problem, \( x \) is one-fourth of its supplement. We can set up the equation \( x = \frac{1}{4}(180 - x) \) and solve for \( x \).

2. For the second question, let the measures of the two complementary angles be \( 4x \) and \( 5x \). Since they are complementary, their sum is \( 90 \) degrees. We can set up the equation \( 4x + 5x = 90 \) and solve for \( x \).

Let the angle be \( x \). The supplement of the angle is given by \( 180 - x \). According to the problem, we have the equation:

\[ x = \frac{1}{4}(180 - x) \]

To solve for \( x \), we rearrange the equation:

\[ x = 45 - 0.25x \]

Adding \( 0.25x \) to both sides gives:

\[ 1.25x = 45 \]

Dividing both sides by \( 1.25 \):

\[ x = 36 \]

Thus, the measure of the angle is \( 36 \) degrees.

Let the two angles be \( 4x \) and \( 5x \). Since they are complementary, we have:

\[ 4x + 5x = 90 \]

This simplifies to:

\[ 9x = 90 \]

Dividing both sides by \( 9 \):

\[ x = 10 \]

Now, we can find the measures of the two angles:

\[ 4x = 4 \times 10 = 40 \quad \text{and} \quad 5x = 5 \times 10 = 50 \]

Thus, the measures of the two angles are \( 40 \) degrees and \( 50 \) degrees.

Final Answer