Problem 36 - Introductory Algebra for College Students

Question

One angle of a triangle is three times as large as another. The measure of the third angle is \(30^{\circ}\) greater than that of the smallest angle. Find the measure of each angle.

Step-by-Step Solution

Answer

The measures of the angles are \(30^{\circ}\), \(90^{\circ}\), and \(60^{\circ}\) respectively.

Step 1: Define the Angles

Let's denote the smallest angle as \(x\). According to the problem, one angle is three times as large as the smallest one, so that angle is \(3x\). The measure of the third angle is \(30^{\circ}\) greater than that of the smallest angle, meaning the third angle is \(x + 30\).

Step 2: Write and Simplify the Equation

Since all angles in a triangle add up to \(180^{\circ}\), we write the equation for the sum of the angles as: \(x + 3x + (x + 30) = 180\). Simplify this to get \(5x + 30 = 180\)

Step 3: Solve the Equation

Subtract 30 from both sides of the equation to isolate the variable term: \(5x = 150\). Finally, divide both sides by 5 to solve for \(x\): \(x = 30^{\circ}\)

Step 4: Find the Measures of the Other Angles

Substitute \(x = 30^{\circ}\) back in to find the measures of the other angles. The second angle is \(3x = 90^{\circ}\) and the third angle is \(x + 30 = 60^{\circ}\).