Problem 31 - Intermediate Algebra

Question

Two angles are complementary, and the measure of one of the angles is \(10^{\circ}\) less than four times the measure of the other angle. Find the measure of each angle.

Step-by-Step Solution

Answer

The angles measure 70 degrees and 20 degrees.

Step 1: Understanding Complements

Complementary angles are two angles whose measures add up to 90 degrees. We are given that two angles, let's call them \(x\) and \(y\), are complementary.

Step 2: Setting Up the Equation

From the problem, we know that the measure of one angle is 10 degrees less than 4 times the measure of the other angle. Let's assume \(x\) is the angle that is 10 degrees less than four times \(y\). So we can write this as an equation: \(x = 4y - 10\).

Step 3: Forming the System of Equations

Since the angles are complementary, their measures add up to 90 degrees. So, we have: \(x + y = 90\). Now we have two equations:1. \(x = 4y - 10\)2. \(x + y = 90\).

Step 4: Substitute and Solve for y

Substitute the expression for \(x\) from the first equation into the second equation: \((4y - 10) + y = 90\). Simplify this equation: \(5y - 10 = 90\). Add 10 to both sides: \(5y = 100\). Now, divide both sides by 5: \(y = 20\).

Step 5: Substitute and Solve for x

Now that we have \(y = 20\), substitute this value back into the equation \(x = 4y - 10\): \(x = 4(20) - 10\). Calculate \(x\): \(x = 80 - 10\), thus \(x = 70\).

Step 6: Verify Solution

Verify the solution by adding the measures of the angles: \(x + y = 70 + 20 = 90\). Since the angles add up to 90 degrees and meet the problem conditions, our solution is correct.

Key Concepts

Understanding Angle Measures

When studying geometry, angle measures play a central role in understanding the properties of shapes and angles. In this exercise, we are dealing with **complementary angles**. These are two angles whose measures add up to 90 degrees. This concept is essential when you need to find unknown angle measures based on a known relationship. For example, if one angle is known to be \(x\) and the other \(y\), and they are complementary, the relationship can be represented as:

• \( x + y = 90^{\circ} \)

Knowing that the sum of these angles is 90 degrees allows you to create equations to find their exact values, especially when additional conditions, such as those described in the exercise, are given.

Algebraic Equations in Angle Problems

Algebraic equations help translate verbal conditions into mathematical statements, making it easier to solve unknowns. In this exercise, the second angle measure is provided through a descriptive condition: "the measure of one of the angles is 10 degrees less than four times the measure of the other angle." Translating this into an algebraic equation, assuming \(x\) is the angle with the given condition and \(y\) is the other angle, gives:

• \( x = 4y - 10 \)

This is a linear equation, a basic yet powerful tool in algebra that links the measures of the angles through the described relationship. Formulating and solving such equations is crucial in finding specific angle measures in geometric problems.

Using Systems of Equations

A system of equations, such as the one in this problem, consists of multiple equations working together to find the values of unknown variables. Here, we have:

• \( x = 4y - 10 \)
• \( x + y = 90 \)

The strategy is to use one of these equations to express one variable (\(x\)) in terms of the other (\(y\)), and substitute this expression into the second equation. This method is known as **substitution**. By solving the equations step-by-step:
- First, substitute \(x\) from the first equation into the second: \((4y - 10) + y = 90\).- Simplify and solve for \(y\): \(5y - 10 = 90\), which simplifies to \(y = 20\) after solving.
Once \(y\) is known, plug it back into the first equation to solve for \(x\): \(x = 4(20) - 10\), resulting in \(x = 70\).
Using systems of equations allows for a systematic approach to solving problems involving multiple variables by isolating and solving for each unknown.