Problem 46 - Introductory Algebra for College Students

Question

Use the five-step problem-solving strategy to find the measure of the angle described. The angle's measure is \(78^{\circ}\) less than that of its complement.

Step-by-Step Solution

Answer

The measure of the angle is \(6^{\circ}\)

Step 1 - Decode the problem and set up an equation

The first step is to understand the problem and set up an equation. We know that in complementary angles, the two angles add up to \(90^{\circ}\). For instance, one angle is \(x\) and its complement will be \(90^{\circ} - x\). According to the problem, the angle is \(78^{\circ}\) less than its complement. So, we can represent it as \(x = 90^{\circ} - x - 78^{\circ}\).

Step 2 - Solve the equation for \(x\)

Now, the equation \(x = 90^{\circ} - x - 78^{\circ}\) should be solved for \(x\). First, combine like terms on the right side of the equation, making it \(x = 12^{\circ} - x\). Then, add \(x\) on both sides to get \(2x = 12^{\circ}\). Lastly, divide both sides by 2 to solve for \(x\). The result is \(x = 6^{\circ}\).

Step 3 - Verify the solution

After finding \(x = 6^{\circ}\), you must verify this solution. The complement of \(6^{\circ}\) is \(90^{\circ} - 6^{\circ} = 84^{\circ}\), which should be \(78^{\circ}\) more than \(x\). Since it is indeed larger than \(6^{\circ}\) by \(78^{\circ}\), \(x = 6^{\circ}\) is confirmed to be the correct solution.