Problem 32 - Introductory Algebra for College Students

Question

Use the relationship among the three angles of any triangle to solve. (image cannot copy) One angle of a triangle is three times as large as another. The measure of the third angle is \(40^{\circ}\) more than that of the smallest angle. Find the measure of each angle.

Step-by-Step Solution

Answer

The measures of the angles are \(28^{\circ}\), \(84^{\circ}\), and \(68^{\circ}\).

Step 1: Identify the variables

Let's call the smallest angle x. The second angle which is three times as large as the smallest would then be 3x. The third angle which is \(40^{\circ}\) more than the smallest angle would be \(x + 40^{\circ}\). These are our three angles.

Step 2: Set up the equation

We know that the sum of the angles of a triangle is \(180^{\circ}\). So, if we add up our three angles we should get \(180^{\circ}\). This gives us the equation \(x + 3x + x + 40^{\circ} = 180^{\circ}\).

Step 3: Simplify the equation

Combine like terms on the left side of the equation. This simplifies the equation to \(5x + 40^{\circ} = 180^{\circ}\).

Step 4: Solve for x

Subtract \(40^{\circ}\) from both sides of the equation to isolate x terms to one side. This results in \(5x = 140^{\circ}\). Then divide both sides by 5 to find the value of x, which gives \(x = 28^{\circ}\). This is the measure of the smallest angle.

Step 5: Find the measure of the other angles

Substitute x = 28 into the expressions for the other two angles to find their measures. The second angle is 3x which gives \(3*28^{\circ} = 84^{\circ}\). The third angle is \(x + 40^{\circ}\) which gives \(28^{\circ} + 40^{\circ} = 68^{\circ}\).

Key Concepts

Angle Relationships in Triangles

Understanding the intricate relationships between angles in triangles is fundamental in geometry. In every triangle, there exist three angles, and central to their relationship is an important property - the internal angles of a triangle always add up to a sum of 180 degrees.

This knowledge allows us to set up relationships and equations when given partial information about a triangle's angles. For instance, if a problem states that one angle is three times another or 40 degrees more than another, these descriptions can be translated into algebraic expressions representing each angle. Once we have these expressions, we can use the principle that the sum of all angles equals 180 degrees to solve for the unknown values.

Algebraic Expression for Angles

An algebraic expression is a mathematical phrase that includes numbers, variables, and operators to represent a certain value. In the context of triangles, we create algebraic expressions to represent the measures of angles.

For example, in the given problem, the smallest angle is represented by the variable x. Other angles are defined in relation to x. The second angle is three times the smallest, represented by the expression 3x. The third angle is expressed as x plus 40 degrees, or x + 40. These expressions help in formulating an equation that encompasses the angle sum property of triangles. By understanding how to translate word problems into algebraic expressions, students can more effectively tackle a variety of geometric problems involving angles.

Triangle Angle Sum Theorem

The Triangle Angle Sum Theorem is a critical concept in geometry. It states that the sum of the interior angles of a triangle is always 180 degrees. This theorem is instrumental when solving for unknown angles.

In applying this theorem to the given exercise, we sum the algebraic expressions of the angles: x (smallest angle), 3x (second angle), and x + 40 (third angle), and set the equation equal to 180 degrees. Simplifying and solving this equation reveals the value of x, and by subsequently plugging this value back into the expressions for the three angles, we find their exact measure.

This theorem not only gives us a straightforward formula for problem-solving but also provides a deeper understanding of the balanced nature of angles within a triangle.