Problem 32 - Prealgebra

Question

ALGEBRA The measures of the angles of a triangle are in the ratio \(1: 4: 5 .\) Find the measure of each angle. (lesson \(9-3\) )

Step-by-Step Solution

Answer

The measures of the angles are \(18°\), \(72°\), and \(90°\).

Step 1: Understand the Sum of Angles in a Triangle

In any triangle, the sum of the interior angles is always 180 degrees. We will use this fact to solve the problem.

Step 2: Define Variables for the Angles Based on the Given Ratio

Define the smallest angle as \( x \). Then, based on the ratio \(1:4:5\), the angles can be expressed as \( x \), \( 4x \), and \( 5x \).

Step 3: Write an Equation for the Sum of Angles

Since the sum of the angles is 180 degrees, we set up the equation: \( x + 4x + 5x = 180 \).

Step 4: Simplify the Equation

Combine like terms in the equation: \( 10x = 180 \).

Step 5: Solve for x

Divide both sides of the equation by 10 to solve for \( x \): \( x = 18 \).

Step 6: Find the Measures of Each Angle

Substitute \( x = 18 \) back into the expressions for each angle: the angles are \( x = 18° \), \( 4x = 72° \), and \( 5x = 90° \).

Step 7: Confirm the Solution is Correct

Add the angles together: \( 18° + 72° + 90° = 180° \). This confirms the solution is correct as it matches the sum of angles in a triangle.

Key Concepts

Ratio of Angles

In a triangle setup, angles can often be in a specific ratio relative to each other. Understanding this concept of ratios helps in predicting the actual measures of angles based on their relational sizes. For instance, if the ratio is given as \(1:4:5\), we understand that one angle is relatively smaller compared to the others. The ratio indicates:

• The first angle is the baseline, \(1x\).
• The second angle is four times larger, \(4x\).
• The third angle is five times larger, \(5x\).

By using a variable, you can easily scale these ratios to find actual angle measures. It's important in numerous geometric scenarios where precise angle determination is crucial.

Sum of Angles in Triangle

All triangles in Euclidean geometry share a crucial trait: their internal angles always add up to 180 degrees. This property is fundamental when solving for unknown angles. Consider that:

• It doesn't matter the triangle type—whether it's acute, obtuse, or right-angled—the rule holds.
• Knowing this property allows you to set equations for unknown angles with confidence.
• It's a cornerstone for proofs and problem-solving in geometry.

For example, when given angle measurements as part of a ratio, simply set up the equation to equal 180 degrees. This will enable you to solve for each angle individually.

Solving Linear Equations

Linear equations are a basic yet powerful tool for finding unknown values. When given relationships, such as angle ratios in a triangle, linear equations enable the calculation of those unknowns. Here's a simple breakdown:

• First, translate the problem into an equation using all known information, like the sum of angles.
• Next, combine like terms to simplify the equation—a typical action being combining variable terms.
• Finally, isolate the variable by performing inverse operations, like division in our triangle example.

With practice, setting up and solving these equations becomes intuitive, facilitating easier solutions to more complex problems over time. Understanding these steps will build a solid foundation for algebraic thinking and problem-solving.