Problem 36 - Introductory and Intermediate Algebra for College Students 4th

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 triangle angles are \(30^{\circ}\), \(90^{\circ}\), and \(60^{\circ}\).

Step 1: Define the Variables

Let's denote the smallest angle as \(x^{\circ}\). Therefore, the angle that is three times larger will be \(3x^{\circ}\) and the third angle will be \(x+30^{\circ}\).

Step 2: Write down the Equation

From the properties of triangles, we know that the sum of the internal angles equals \(180^{\circ}\). So, the equation turns into \(x + 3x + x + 30 = 180\).

Step 3: Simplify the Equation

Summing up the x coefficients, the equation simplifies to \(5x + 30 = 180\).

Step 4: Solve for x

Subtract 30 from both sides and then divide both sides by 5. It results in \(x = 30^{\circ}\).

Step 5: Find the Measures of All Angles

Substitute \(x = 30^{\circ}\) back into the initial definitions. Thus, the three angles will be \(30^{\circ}\), \(90^{\circ}\), and \(60^{\circ}\).

Key Concepts

Algebraic Equations

In this problem, using algebraic equations helps us find unknown angles in a triangle. We began by defining a variable for the smallest angle. By letting the smallest angle be represented as \(x\), we created a formula for each angle:

• Smallest angle: \(x\)
• Second angle: \(3x\)
• Third angle: \(x + 30\)

Algebraic equations are useful in transforming such a problem involving angles into a solvable mathematical equation. Remember, an equation is just like a mathematical sentence, where all parts must balance. Our equation derived from the triangle's angle properties is:\[x + 3x + x + 30 = 180\]This representation illustrates how unknown quantities can relate and eventually allows us to pinpoint their exact values through calculation.

Geometry

Geometry plays a central role in solving problems involving shapes and angles, such as triangles. A unique aspect of every triangle is that the sum of its internal angles is always \(180^{\circ}\). This property serves as the foundation for setting up our equation. Each angle defined algebraically contributes to this total, thus connecting algebraic understanding with geometric principles.
The structure of our specific triangle is based upon the geometric relations established:

• One angle is thrice another, which implies a ratio relationship.
• The third angle adjusts itself, being \(30^{\circ}\) greater than the smallest angle.

Effectively using geometry to outline these relationships allows us to confidently apply algebra to our framework, helping us reach the solution.

Problem Solving

Problem-solving is a step-by-step approach to unraveling questions or challenges. Here, we tackled the triangle problem by dissecting it into manageable parts. The steps included understanding the problem, defining variables, setting up equations, and solving them systematically:

• Understanding the Problem: Recognize the properties of triangle angles and their given relationships.
• Defining the Variables: Use logical designations (e.g., let \(x\) represent the smallest angle) to indicate unknowns.
• Setting Up the Equation: Integrate both algebraic equations and geometric rules into a coherent equation.
• Solving the Equation: Follow logical simplifications to isolate and compute the unknown values.

These problem-solving steps are vital in decoding math problems, systematically ensuring that each part aligns with the given properties and conditions. Aligning each stage methodically can make solving complex problems a smoother process.