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 angles are \(18^{\circ}\), \(72^{\circ}\), and \(90^{\circ}\).

Step 1: Express Angles in Terms of a Single Variable

Let's assume the measures of the angles are \(x\), \(4x\), and \(5x\) respectively, since their ratio is given as \(1: 4: 5\).

Step 2: Set Up the Equation Using the Angle Sum

We know that the sum of the angles in any triangle is \(180^{\circ}\). Therefore, we can write the equation: \(x + 4x + 5x = 180^{\circ}\).

Step 3: Simplify the Equation

Combine the terms on the left-hand side: \(10x = 180\).

Step 4: Solve for x

To find \(x\), divide both sides of the equation by \(10\): \(x = \frac{180}{10} = 18\).

Step 5: Find the Measure of Each Angle

Now that we know \(x = 18\), we can find the measures of the angles: \(18^{\circ}\), \(4 \times 18^{\circ} = 72^{\circ}\), and \(5 \times 18^{\circ} = 90^{\circ}\).

Key Concepts

Ratios in Mathematics

Ratios are a fundamental concept in mathematics that involve comparing two or more quantities. When examining angles in a triangle, a ratio denotes the relative size of each angle compared to others. In this exercise, we have a ratio of \(1:4:5\), which helps us understand how one angle relates to another.

To work with ratios, we assign a variable to represent a base unit. For instance, if the smallest angle is \(x\), the others are proportionally \(4x\) and \(5x\). This technique of using a single variable simplifies calculations and can be very helpful in solving problems where measures or lengths follow determined ratios.

Here are some tips for working with ratios effectively:

• Identify the quantities you’re comparing and express their relationship through a simple fraction or ratio.
• Find a common variable that represents the simplest form of the ratio value.
• Use algebraic methods to solve for the unknown variable when relevant.

Understanding ratios allows you to solve a range of mathematical puzzles by setting a proportional relationship between different elements.

Solving Equations

Solving equations is a key mathematical skill, crucial for finding unknown variables within an expression. In this case, the aim is to find the value of \(x\) from the angle sum equation of a triangle. Let's go through the essential steps:

First, we set up an equation using the property that the sum of angles in a triangle is always \(180^{\circ}\). Here’s how it works:

• Write the equation according to our ratios: \(x + 4x + 5x = 180\).
• Combine like terms to simplify: We end up with \(10x = 180\).

Next comes finding the value of \(x\). Divide both sides by \(10\) to solve for \(x\), yielding \(x = 18\). This process of solving equations by simplifying and isolating variables forms the backbone of algebra and allows us to find individual angle measures when substituted back into the ratio.

Remember:

• Always check that the solution is consistent with the given problem conditions.
• Verify your work by substituting back into the original equation or conditions.

Properties of Triangles

Understanding the properties of triangles can make these kinds of problems much easier. Every triangle has certain properties that are always true, such as the sum of interior angles equaling \(180^{\circ}\). These properties serve as the foundation for solving many geometric problems.

The Angle Sum Property is pivotal in this exercise. It states:

• The sum of all angles in a triangle is invariably \(180^{\circ}\).

By using this property, you can create an equation that helps find unknown angles or side lengths when certain conditions are provided. This concept is not just limited to angle measures.It also applies to understand other properties, like side lengths and types of triangles like equilateral, isosceles, and scalene.

Here are other regular triangle characteristics to remember:

• A triangle's exterior angle is equal to the sum of the opposite interior angles.
• The longest side is opposite the largest angle, and the shortest side is opposite the smallest angle.

By understanding these key properties, students can more easily solve triangle-related mathematical problems and apply these insights to various geometrical challenges.