Question
An angle measures 54° more than the measure of its complementary angle. What is the measure of each angle?
Answers
GPT-4o mini
Let the measure of the unknown angle be \( x \) degrees. The complementary angle, which is \( 90^\circ - x \), is defined as the angle that, when added to \( x \), equals \( 90^\circ \).
According to the problem, we can express the relationship between the angles as follows:
\[
x = (90^\circ - x) + 54^\circ
\]
Now, we can simplify this equation:
\[
x = 90^\circ - x + 54^\circ
\]
Combine like terms:
\[
x = 144^\circ - x
\]
Now, add \( x \) to both sides:
\[
x + x = 144^\circ
\]
\[
2x = 144^\circ
\]
Next, divide both sides by 2:
\[
x = 72^\circ
\]
Now that we have the measure of the unknown angle, we can find its complementary angle:
\[
90^\circ - x = 90^\circ - 72^\circ = 18^\circ
\]
Thus, the measures of the angles are:
- The unknown angle: \( 72^\circ \)
- The complementary angle: \( 18^\circ \)
To confirm, we check that \( 72^\circ \) is indeed \( 54^\circ \) more than \( 18^\circ \):
\[
18^\circ + 54^\circ = 72^\circ
\]
This is correct. Therefore, the measures of the angles are:
- **72°**
- **18°**
According to the problem, we can express the relationship between the angles as follows:
\[
x = (90^\circ - x) + 54^\circ
\]
Now, we can simplify this equation:
\[
x = 90^\circ - x + 54^\circ
\]
Combine like terms:
\[
x = 144^\circ - x
\]
Now, add \( x \) to both sides:
\[
x + x = 144^\circ
\]
\[
2x = 144^\circ
\]
Next, divide both sides by 2:
\[
x = 72^\circ
\]
Now that we have the measure of the unknown angle, we can find its complementary angle:
\[
90^\circ - x = 90^\circ - 72^\circ = 18^\circ
\]
Thus, the measures of the angles are:
- The unknown angle: \( 72^\circ \)
- The complementary angle: \( 18^\circ \)
To confirm, we check that \( 72^\circ \) is indeed \( 54^\circ \) more than \( 18^\circ \):
\[
18^\circ + 54^\circ = 72^\circ
\]
This is correct. Therefore, the measures of the angles are:
- **72°**
- **18°**