Question

If a supplement of an angle has a measure 78 less than the measure of the angle, what are the measures of the angles?

Answer 1

Let's denote the measure of the angle as \( x \). The supplement of an angle is given by \( 180^\circ - x \).

According to the problem, the supplement of the angle measures 78 degrees less than the angle itself. We can set up the following equation:

\[
180^\circ - x = x - 78^\circ
\]

Now, we can solve this equation for \( x \):

1. Add \( x \) to both sides:

\[
180^\circ = 2x - 78^\circ
\]

2. Add \( 78^\circ \) to both sides:

\[
180^\circ + 78^\circ = 2x
\]

\[
258^\circ = 2x
\]

3. Divide both sides by 2:

\[
x = \frac{258^\circ}{2} = 129^\circ
\]

Now that we have the measure of the angle, we can find its supplement:

\[
180^\circ - x = 180^\circ - 129^\circ = 51^\circ
\]

Thus, the measures of the angles are:

- The angle: \( 129^\circ \)
- Its supplement: \( 51^\circ \)

To verify, check if the supplement (51 degrees) is indeed 78 degrees less than the angle (129 degrees):

\[
129^\circ - 78^\circ = 51^\circ
\]

This confirms our solution. Therefore, the measures of the angles are:

- Angle: \( 129^\circ \)
- Supplement: \( 51^\circ \)