Problem 47 - Algebra for College Students

Question

Set up an algebraic equation and solve each problem. One angle of a triangle has a measure of \(60^{\circ}\) and the measures of the other two angles are in the ratio of 2 to 3. Find the measures of the other two angles.

Step-by-Step Solution

Answer

The other two angles measure 48° and 72°.

Step 1: Understand the Problem

We are given that one angle of a triangle is \(60^{\circ}\). The other two angles are in the ratio of 2:3. We need to find the measures of these two angles.

Step 2: Set Up the Equation

Let the measures of the other two angles be \(2x\) and \(3x\). Since the sum of angles in a triangle is \(180^{\circ}\), we can set up the equation: \[60 + 2x + 3x = 180\]

Step 3: Simplify the Equation

Combine like terms in the equation: \[60 + 5x = 180\]

Step 4: Isolate the Variable

Subtract 60 from both sides to solve for \(x\): \[5x = 120\]

Step 5: Solve for x

Divide both sides by 5 to find \(x\): \[x = 24\]

Step 6: Find the Measures of the Angles

Substitute \(x\) back into the expressions for the angles to find their measures: \(2x = 2 \times 24 = 48^{\circ}\) and \(3x = 3 \times 24 = 72^{\circ}\).

Key Concepts

Algebraic Equations

Algebraic equations are mathematical sentences that use variables to represent numbers. They help us find unknown values using given information and relationships. In the context of triangle angle relationships, algebraic equations can be powerful tools. For example, if an equation represents the angles of a triangle, such as

• \[60 + 2x + 3x = 180\]

we begin with known values (like one angle being \(60^{\circ}\)) and unknowns (like \(x\)). Here, variables \(2x\) and \(3x\) are used for the unknown angles. Simply put, algebraic equations help us translate a word problem into a solvable mathematical format. This conversion is crucial as it lays the groundwork to find the solution efficiently.

Ratio Problems

Ratios express a relationship between two or more quantities, showing how many times one value contains another. When solving ratio problems in triangles, it is common to use expressions like \(2x\) and \(3x\) if angles are in a 2:3 ratio. This sort of problem often appears within constraints, such as the fixed sum of angles in a triangle.

• Think of the ratio 2:3 as a scale. It tells us that one part is 2 times a unit \(x\) and the other is 3 times a unit \(x\).
• Apply this ratio by expressing the angles as \(2x\) and \(3x\).

Understanding ratios can simplify complex problems by establishing proportionality, making it easier to set up equations for unknowns.

Sum of Angles in a Triangle

The sum of angles in a triangle is a fundamental concept in geometry. Every triangle, regardless of its type, has interior angles that always add up to \(180^{\circ}\). This rule is essential for solving problems involving triangles, including finding unknown angles.

• Knowing the sum is \(180^{\circ}\) allows us to set up equations quickly. For instance, one angle given as \(60^{\circ}\), lets us write:\[60 + 2x + 3x = 180\]
• By solving this equation, we find the other two angles.

This concept provides a consistent method to check your work, ensuring the calculated angles do add up to \(180^{\circ}\). It is one of the simplest yet most useful tools in geometry for verifying the accuracy of your solutions.