Question

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

Answer 1

Step 1: Defining the angles:

Let the smaller angle be \( x \). Then, the larger angle is \( x + 18^\circ \), as stated in the problem.

Step 2: Using the supplementary angle property:

Since the two angles are supplementary, their sum is \( 180^\circ \). Therefore, we can write the equation:
\[ x + (x + 18^\circ) = 180^\circ \]

Step 3: Simplifying the equation:

Simplifying the left-hand side:
\[ x + x + 18^\circ = 180^\circ \] \[ 2x + 18^\circ = 180^\circ \] Now, subtract \( 18^\circ \) from both sides:
\[ 2x = 162^\circ \] Next, divide both sides by 2:
\[ x = \frac{162^\circ}{2} = 81^\circ \]

Step 4: Finding the larger angle:

Since the smaller angle is \( 81^\circ \), the larger angle is:
\[ 81^\circ + 18^\circ = 99^\circ \]

Step 5: Conclusion:

Thus, the smaller angle is \( 81^\circ \), and the larger angle is \( 99^\circ \).