Problem 207 - Elementary Algebra

Question

In the following exercises, translate to a system of equations and solve. The difference of two complementary angles is 30 degrees. Find the measures of the angles.

Step-by-Step Solution

Answer

The measures of the angles are 60 degrees and 30 degrees.

Step 1 - Define the variables

Let the two complementary angles be denoted as \(x\) and \(y\). Complementary angles sum up to 90 degrees.

Step 2 - Set up the equations

Use the given information to set up two equations: 1) The sum of the angles: \(x + y = 90\)2) The difference of the angles: \(x - y = 30\)

Step 3 - Solve the system of equations using addition

Add the two equations: \(x + y + (x - y) = 90 + 30\)This simplifies to:\(2x = 120\)\(x = 60\)

Step 4 - Substitute back to find the second angle

Use the value of \(x\) to find \(y\): \(60 + y = 90\)\(y = 90 - 60\)\(y = 30\)

Step 5 - State the final measures of the angles

The two angles are 60 degrees and 30 degrees.

Key Concepts

complementary angles

Complementary angles are two angles whose measures add up to 90 degrees. This is a key concept in geometry used to solve many problems involving angle measures. For example, if one angle is known, you can always find the other angle by subtracting the known angle from 90 degrees. To make it more relatable, imagine a right angle (which is 90 degrees) being split into two smaller angles. These smaller angles are complementary because together they form the right angle.

Here's a quick tip: if you ever forget what complementary means, just remember that the word 'complementary' starts with 'C', and 'C' is the third letter of the alphabet. The sum of two complementary angles is always the third letter squared, i.e., 90 degrees!

systems of equations

A system of equations is when you have two or more equations that share the same variables. In this problem, we have two equations that relate the two complementary angles:

1) The sum of the angles: \( x + y = 90 \)
2) The difference of the angles: \( x - y = 30 \)

To solve the system of equations, we use methods like substitution, elimination, or graphical representation. In this solution, we used the elimination method. We added the two equations to eliminate one of the variables and find the value of the other. It’s like having two pieces of a puzzle that you need to fit together—knowing one part helps you find the other.

Solving systems of equations helps you find the value of each variable in a structured manner. It is widely used in various fields like physics, engineering, and economics.

angle measures

Understanding how to measure and calculate angles is fundamental to geometry. Angle measures are typically given in degrees. In this exercise, we calculated the measures of two unknown angles which were constrained by the rules of complementary angles.

First, we defined our angles as \( x \) and \( y \). Since they are complementary, their sum is 90 degrees. The given problem also mentioned that their difference is 30 degrees. Using this information, we formed two equations:

1) \( x + y = 90 \)
2) \( x - y = 30 \)

By solving these equations step-by-step, we discovered that one angle is 60 degrees and the other is 30 degrees. This approach ensures that both conditions of the problem are satisfied.

Getting comfortable with angle measures and how to manipulate them mathematically can simplify the process of solving more complex geometrical problems. It all starts with clearly understanding what the problem is asking for and then methodically breaking it down using equations.