Problem 38 - Algebra A Combined Function

Question

Find the measures of two supplementary angles if one angle is \(18^{\circ}\) more than twice the other.

Step-by-Step Solution

Answer

The two angles measure \(54^{\circ}\) and \(126^{\circ}\).

Step 1: Define the Variables

Let the measure of the first angle be \(x\). Since the first angle is \(x\), the second angle, which is described as being \(18^{\circ}\) more than twice the first angle, can be represented as \(2x + 18^{\circ}\).

Step 2: Use the Supplementary Angle Relationship

Supplementary angles add up to \(180^{\circ}\). Therefore, we can set up the equation: \(x + (2x + 18^{\circ}) = 180^{\circ}\).

Step 3: Simplify the Equation

Combine like terms in the equation from Step 2: \(x + 2x + 18 = 180\) This simplifies to: \(3x + 18 = 180\).

Step 4: Solve for x

Subtract \(18\) from both sides to isolate the terms involving \(x\): \(3x = 162\).Now, divide by \(3\) to solve for \(x\): \(x = 54\).

Step 5: Solve for the Second Angle

Substitute \(x = 54\) back into the expression for the second angle: \(2x + 18\) \(2(54) + 18 = 108 + 18 = 126\).So, the second angle is \(126^{\circ}\).

Step 6: Verify the Solution

Check that the sum of the two angles is \(180^{\circ}\): \(54 + 126 = 180\).Thus, both angles indeed sum up to \(180^{\circ}\), confirming the solution is correct.

Key Concepts

Angle Relationships

Understanding angle relationships is key to solving problems involving supplementary angles. Supplementary angles are two angles whose measures add up to exactly \(180^{\circ}\). This concept is crucial in geometry because it helps you determine unknown angle measures when given certain information about their relationships.
For example, if one angle is given, you can find the supplementary angle by subtracting the given angle from \(180^{\circ}\). Knowing angle relationships also aids in visualizing and solving real-world problems, such as finding angles formed by intersecting lines and corners in various objects.
To better grasp this concept, consider these key points:

• Supplementary angles sum to \(180^{\circ}\).
• If one angle is known, the other can be found by subtracting it from \(180^{\circ}\).
• This relationship is widely applied in triangle exterior angles and linear pairs.

Understanding angle relationships helps build a strong foundation in geometry, enabling you to tackle various mathematical problems with confidence.

Algebraic Equations

Algebraic equations are expressions that contain variables and constants, structured to be equal to a number or another expression. In problems involving supplementary angles, algebraic equations help us find unknown angle measures.
For the given problem, you define the first angle as \(x\), and the second angle becomes \(2x + 18^{\circ}\) based on the information provided. Then, by using the knowledge of supplementary angles, you can set up the equation \(x + (2x + 18^{\circ}) = 180^{\circ}\).
This type of problem teaches you how to translate real-world situations into mathematical expressions and equations. Some key takeaways include:

• Define variables clearly based on the information given.
• Establish relationships between unknowns using equations.
• Solve equations by simplifying and isolating the variable.

Mastering algebraic equations is essential for solving a wide range of mathematical problems efficiently.

Mathematical Problem-Solving

Mathematical problem-solving involves using logical reasoning, critical thinking, and mathematical concepts to find solutions. In the example problem, solving for two supplementary angles requires a step-by-step approach to arrive at the correct answer.
The problem-solving process begins with interpreting the problem and translating it into mathematical expressions, as seen with defining the variables and setting up the equation. This is followed by simplifying and solving the equation to find unknown values.
Effective problem-solving requires:

• Understanding the problem statement clearly.
• Identifying and employing appropriate strategies.
• Applying logical reasoning to job through each step methodically.
• Verifying all solutions for accuracy and completeness.

Developing strong problem-solving skills is invaluable, as it applies beyond academics, aiding in decision-making and logical thinking in everyday life.

Basic Algebra Concepts

Basic algebra concepts form the foundation of solving equations and understanding relationships between variables. In this scenario, these concepts help to find the unknown angles which satisfy the conditions given.
Here are some core algebra concepts highlighted in the problem:

• Variables act as placeholders for unknown numbers, which allows for flexibility in setting up equations.
• Equations are central to representing mathematical relationships. In this case, the equation \(3x + 18 = 180\) is derived and then solved to find \(x\).
• Simplification involves combining like terms to streamline the equation, making it easier to solve.
• Operations like addition, subtraction, multiplication, and division are used to isolate the variable and find its value.

Grasping these basic algebra concepts is essential for solving a multitude of problems effectively and lays the groundwork for more advanced mathematical study.