Problem 35 - Algebra A Combined Function

Question

Recall that two angles are complementary if the sum of their measures is \(90^{\circ} .\) Find the measures of two complementary angles if one angle is twice the other.

Step-by-Step Solution

Answer

The angles are \(30^{\circ}\) and \(60^{\circ}\).

Step 1: Define Variables

Let one angle be \(x\). Since the other angle is twice the first angle, let it be \(2x\).

Step 2: Set Up the Equation

Since the angles are complementary, their sum is \(90^{\circ}\). Therefore, you can write the equation: \[ x + 2x = 90^{\circ} \]

Step 3: Solve for x

Simplify the equation to find \(x\): \[ 3x = 90^{\circ} \] Divide both sides by 3: \[ x = 30^{\circ} \]

Step 4: Determine Both Angles

Now that \(x = 30^{\circ}\), find the measure of the other angle which is \(2x\): \[ 2x = 2 \times 30^{\circ} = 60^{\circ} \]

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that involve variables and constants. They show the relationship between different quantities. In this problem, the equation is all about the angle measures. The key equation derived is: \[ x + 2x = 90^{\circ} \] This equation helps us express the relationship between the complementary angles. Since these angles add up to 90 degrees, the equation captures this condition. By solving this equation, we can easily find the measure of each angle. In general, equations like these allow us to solve various real-world problems by finding unknown values.

Variable Definition

Defining variables is a crucial step in solving any mathematical problem. In this exercise, we use variables to represent unknown quantities.

• Let \(x\) represent the measure of the first angle.
• Let \(2x\) represent the measure of the second angle which is twice the first.

By defining these variables, you establish a clear foundation for setting up your equation. It's important to choose variables that logically fit the problem scenario. Proper variable definition simplifies understanding and solving the problem.

Angle Measures

In geometry, understanding angle measures is fundamental. Complementary angles are two angles whose measures add up to 90 degrees. Here, we are dealing with two specific angles:

• The first angle is \(x = 30^{\circ}\).
• The second angle, being twice the first, is \(2x = 60^{\circ}\).

Knowing how to measure and calculate angles, especially complementary ones, is essential for grasping more complex geometric concepts. Remember, the sum of 90 degrees in complementary angles ensures that they form a right angle together.

Problem Solving in Mathematics

Problem-solving in mathematics often involves breaking down a problem into manageable steps. Here's how you approach this problem:
1. **Understand the Problem**: Identify what is given and what needs to be found. 2. **Define Variables**: Choose variables to represent unknown values. 3. **Formulate an Equation**: Develop an equation that relates the variables. 4. **Solve the Equation**: Simplify and calculate to find the values of the variables. 5. **Verify the Solution**: Check if the found values satisfy the problem's conditions.
These steps are repeatable and can help you tackle a variety of mathematical problems. Developing these skills is crucial for success in mathematics and related fields.