Problem 39 - Algebra A Combined Function

Question

Two angles are supplementary if their sum is \(180^{\circ} .\) The larger angle below measures eight degrees more than three times the measure of the smaller angle. If \(x\) represents the measure of the smaller angle and these two angles are supplementary, find the measure of each angle.

Step-by-Step Solution

Answer

The smaller angle measures \(43^\circ\) and the larger angle measures \(137^\circ\).

Step 1: Define the Variables

Let the measure of the smaller angle be represented by \(x\). Then, according to the problem, the larger angle is eight degrees more than three times the smaller angle. Therefore, express the larger angle as \(3x + 8\).

Step 2: Set Up the Equation

Since the two angles are supplementary, their sum must equal \(180^\circ\). Set up the equation: \(x + (3x + 8) = 180\).

Step 3: Simplify the Equation

Combine like terms within the equation: \(x + 3x + 8 = 180\) simplifies to \(4x + 8 = 180\).

Step 4: Solve for x

Subtract 8 from both sides to isolate the terms with \(x\): \(4x = 172\). Divide both sides by 4 to solve for \(x\): \(x = 43\). This means the measure of the smaller angle is \(43^\circ\).

Step 5: Find the Measure of the Larger Angle

Substitute \(x = 43\) back into the expression for the larger angle: \(3(43) + 8 = 129 + 8 = 137\). Thus, the measure of the larger angle is \(137^\circ\).

Key Concepts

Angle Measurement

Angles are a fundamental element in geometry and play a key role in understanding various geometric shapes and structures.
A full circle measures 360 degrees, and angles are typically measured in degrees. When two lines intersect, they create an angle that can be acutely sharp, right, obtuse, or any combination that makes up supplementary angles.
Supplementary angles are a pair of angles whose measures add up to exactly 180 degrees. For example:

• If one angle measures 110 degrees, the other must measure 70 degrees because 110 + 70 = 180.
• If one angle is labeled as "x", the supplementary angle can be calculated by finding "180 - x".

Understanding how to measure and calculate angles is crucial because it forms the foundation for more complex geometric calculations. Hence, knowing what supplementary angles are and being able to find the measure of unknown angles are essential skills for solving geometric problems.

Linear Equations

Linear equations are equations of the first order, meaning they involve variables raised to the power of one. They have a standard form, usually written as \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.
In cases with geometry problems like our exercise, linear equations are often used to establish relationships between unknown variables. For example, when trying to find the dimensions of two supplementary angles, linear equations help express one angle in terms of the other.
This is crucial because:

• Writing an equation in terms of known relationships, like the relationship between supplementary angles, allows us to model and solve real-world problems.
• Linear equations simplify these relationships into manageable algebraic expressions that can be solved systematically.

By converting word problems into linear equations, such as finding angles in the exercise, we apply algebraic methods to solve for unknown values effectively.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. This is an essential aspect of algebra and is central to solving many geometric problems.
The process typically involves:

• Combining like terms to simplify the equation. For instance, from our exercise, combining "\(x + 3x + 8 = 180\)" gives "\(4x + 8 = 180\)".
• Isolating the variable by performing operations such as subtraction, addition, multiplication, or division on both sides. So, subtracting 8 from both sides leads to "\(4x = 172\)" followed by dividing both sides by 4 to get "\(x = 43\)".
• Substituting the known values back into other expressions to solve for additional unknowns. For example, substitute \(x = 43\) into "\(3x + 8\)" to find the larger angle.

Understanding how to solve these equations enables students to tackle similar algebraic challenges efficiently, making the previously intimidating problems approachable and solvable.