Problem 50 - Introductory Algebra for College Students

Question

Use the five-step problem-solving strategy to find the measure of the angle described. The measure of the angle's supplement is \(52^{\circ}\) more than twice that of its complement.

Step-by-Step Solution

Answer

The measure of the angle is \(52^{\circ}\).

Step 1: Setting up the equation

Let's denote the angle we are looking for by x. According to the problem, the supplement of the angle (which is \(180^{\circ} - x\)) is 52 degrees more than twice its complement (which is \(90^{\circ} - x\)). We can express this as an equation: \(180^{\circ} - x = 2 * (90^{\circ} - x) + 52^{\circ}\).

Step 2: Simplifying the equation

When we expand the right side of the equation, we end up with: \(180^{\circ} - x = 180^{\circ} - 2x + 52^{\circ}\). We can then simplify the equation by canceling out the \(180^{\circ}\) on both sides: \(-x = -2x + 52^{\circ}\).

Step 3: Solving the equation

Adding 2x to both sides of the equation gives: \(x = 52^{\circ}\). Hence, the measure of the angle is \(52^{\circ}\).

Step 4: Verify the solution

We can validate the answer by checking if the supplement of the angle (\(180^{\circ} - 52^{\circ} = 128^{\circ}\)) is indeed 52 degrees more than twice its complement (which is \(90^{\circ} - 52^{\circ} = 38^{\circ}\), and twice this is \(76^{\circ}\). Adding 52 to 76 gives us 128 degrees, thus confirming our solution is correct.