Question

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

Answer 1

Step 1: Understanding the problem:
We are given that two angles are supplementary, meaning their sum is \(180^\circ\). Also, the greater angle exceeds the smaller one by \(18^\circ\). Let the smaller angle be \(x\), and the greater angle be \(x + 18^\circ\).

Step 2: Set up the equation:
Since the angles are supplementary, we know that their sum is \(180^\circ\):
\[ x + (x + 18^\circ) = 180^\circ \]
Simplify the equation:
\[ 2x + 18^\circ = 180^\circ \]

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

Step 4: Find the greater angle:
The greater angle is \(x + 18^\circ\):
\[ x + 18^\circ = 81^\circ + 18^\circ = 99^\circ \]

Step 5: Conclusion:
The two angles are:
- The smaller angle: \(81^\circ\)
- The greater angle: \(99^\circ\)