Question

The greater of two supplementary angles exceeds the smaller by \(18^\circ\). Find the measures of these two angles.

Answer 1

Step 1: Problem Definition:
Given two supplementary angles where the larger angle exceeds the smaller one by \(18^\circ\). Let the smaller angle be \(x\), and the larger angle be \(x + 18^\circ\).

Step 2: Equation Formulation:
Supplementary angles sum to \(180^\circ\). Therefore:
\[x + (x + 18^\circ) = 180^\circ\]
Simplified equation:
\[2x + 18^\circ = 180^\circ\]

Step 3: Solving for \(x\):
Subtract \(18^\circ\) from both sides:
\[2x = 180^\circ - 18^\circ\]
\[2x = 162^\circ\]
Divide by 2:
\[x = \frac{162^\circ}{2} = 81^\circ\]

Step 4: Determining the Larger Angle:
The larger angle is \(x + 18^\circ\):
\[x + 18^\circ = 81^\circ + 18^\circ = 99^\circ\]

Step 5: Final Angles:
The two angles are:
- Smaller angle: \(81^\circ\)
- Larger angle: \(99^\circ\)