Problem 28 - Prealgebra

Question

One angle in a triangle is half the largest angle, but three times the smallest. Find all three angles.

Step-by-Step Solution

Answer

The smallest angle is 18°, the middle angle is 54°, and the largest angle is 108°.

Step 1: Understand the Problem

We are given a triangle with three angles. Let the smallest angle be \( x \). According to the problem, the middle angle is three times the smallest angle, and the largest angle is twice the middle angle. We need to find all three angles.

Step 2: Assign Variables

Assign variables to the angles. Let the smallest angle be \( x \), the middle angle be \( 3x \), and the largest angle be \( 2 \times 3x = 6x \).

Step 3: Use Triangle Angle Sum Property

The sum of angles in a triangle is always 180°. Therefore, we can write the equation: \( x + 3x + 6x = 180° \).

Step 4: Simplify the Equation

Combine the terms to simplify the equation: \( x + 3x + 6x = 10x = 180° \).

Step 5: Solve for x

Divide both sides by 10 to isolate \( x \): \( x = \frac{180}{10} = 18° \).

Step 6: Calculate All Angles

Now that \( x = 18° \), substitute back to find all angles: Smallest angle is \( x = 18° \), middle angle is \( 3x = 54° \), and the largest angle is \( 6x = 108° \).

Step 7: Verify the Solution

Check that the sum of the angles equals 180°: \( 18° + 54° + 108° = 180° \). The solution satisfies the triangle angle sum property, confirming the angles are correct.

Key Concepts

Angle Sum Property

In a triangle, the sum of the internal angles is one of those everlasting truths of geometry. This is known as the "Angle Sum Property." No matter how large or small a triangle is, the total measure of its three interior angles will always add up to 180°. This principle serves as a fundamental building block for solving many geometric problems.

Understanding this concept is crucial because it allows us to find missing angles when we know at least two angles of a triangle. By applying this property, we start laying down a mathematical framework to delve deeper into the triangle's geometry.

• It can be used to solve equations involving angles.
• It provides stability and simplicity to geometric calculations.
• It serves as a prerequisite for more advanced geometry topics.

For example, in our problem, the angle sum property allows us to write the equation: \( x + 3x + 6x = 180° \), which becomes the foundation of finding each angle's measurement.

Variable Assignment in Math

In mathematical problem-solving, assigning variables to unknown quantities is a technique that allows us to transform real-world problems into mathematical equations.

Variables act as placeholders for these unknown values, enabling the creation of equations that can be solved systematically. They are particularly useful in algebra and pre-algebra contexts, where we deal with unknowns frequently.

• Variables provide a way to represent quantities that might change or that we need to discover.
• This involves giving unknown values with a letter symbol (e.g., \( x \)).
• It simplifies complex problems into more manageable expressions.

In our triangle problem, we assigned \( x \) to the smallest angle. From there, we expressed the other angles in terms of \( x \), as \( 3x \) and \( 6x \). This formed the basis for setting up our critical equation using the angle sum property.

Solving Equations

Solving equations is a cornerstone of algebra and involves finding the value of the variable that makes the equation true. In our problem, we used the equation \( x + 3x + 6x = 180° \) to figure out the unknown angles in the triangle.

The steps to solving an equation typically include:

• Simplifying the equation by combining like terms.
• Isolating the variable using algebraic operations (addition, subtraction, multiplication, or division).
• Checking the solution by substituting back into the original equation.

In this scenario, by combining terms to get \( 10x = 180° \), we simplified the equation enough to solve for \( x \) by dividing both sides by 10. This basic pre-algebra solution path is a stepping stone for more complex algebraic problem-solving scenarios.

Prealgebra Problem Solving

Prealgebra provides foundational skills necessary for understanding more advanced math concepts. It introduces problem-solving methods that are essential for tackling algebraic expressions, equations, and inequalities.

Prealgebra involves learning about various types of numbers and how to manipulate these numbers under different operations. This may include working with whole numbers, fractions, decimals, and even simple equations and inequalities.

• It involves identifying and understanding when and how to use specific mathematical principles.
• Helps develop logical reasoning and critical thinking.
• Prepares students for the transition to formal algebra and beyond.

In our triangle problem, solving such an equation by assigning variables, employing the angle sum property, and equational solving are all quintessential prealgebra skills. Mastery of these skills allows students to approach mathematical problems with confidence and clarity.