Problem 88 - Algebra A Combined Function

Question

Two angles are said to be supplementary if the sum of their measures is \(180^{\circ} .\) If one angle measures \(\frac{x+2}{x}\) degrees, find the measure of its supplement.

Step-by-Step Solution

Answer

The measure of the supplementary angle is \(179 - \frac{2}{x}\) degrees.

Step 1: Understanding Supplementary Angles

Two angles are supplementary if the sum of their measures equals \(180^{\circ}\). This means if you add the two angles together, the total should be \(180^{\circ}\).

Step 2: Define Second Angle

Let the measure of the second angle be \(y\) degrees. Since the two angles are supplementary and one of them is given as \(\frac{x+2}{x}\) degrees, we set up the equation:\[ \frac{x+2}{x} + y = 180 \]

Step 3: Solve for the Supplementary Angle

To find \(y\), rearrange the equation:\[ y = 180 - \frac{x+2}{x} \] This expression gives the measure of the supplementary angle based on the value of \(x\).

Step 4: Simplify the Expression

Rewrite the expression for \(y\):\[ y = 180 - \frac{x+2}{x} = 180 - 1 - \frac{2}{x} = 179 - \frac{2}{x} \]This simplifies further as \(y = 179 - \frac{2}{x}\).

Key Concepts

Angle Measurement

Angles play a crucial role in geometry, from forming shapes to solving complex problems. Measuring angles is all about determining the amount of rotation from one side of the angle to the other. The unit commonly used for this is degrees. One complete rotation forms an angle of 360 degrees. When we talk about angle measures, we can have angles ranging from 0 to 360 degrees.

Supplementary angles are a specific pair where the sum of their measures is exactly 180 degrees. This concept is essential because it helps in identifying linear pairs and understanding angle relationships within various geometric shapes. If you know one angle measure, you can easily find its supplementary partner by subtracting from 180 degrees. This basic concept forms the foundation for solving more complicated geometric problems.

Equation Solving

Solving equations is like solving puzzles. You are given some information (the equation) and must find unknown values that satisfy it. In our exercise, we have an equation involving supplementary angles:

• The first angle is represented as \( \frac{x+2}{x} \) degrees.
• The supplementary angle is denoted by \( y \).
• The two must add up to 180 degrees.

This information sets up our equation: \( \frac{x+2}{x} + y = 180 \). Solving for \( y \) involves isolating it on one side of the equation. This means rearranging the equation gives us \( y = 180 - \frac{x+2}{x} \).

Understanding how to manipulate equations is a crucial mathematical skill. It helps you to express unknown values in terms of known variables, which is an essential tool in both basic and advanced math.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols. They represent a specific value when numbers are substituted for the variables. In our problem, we have an expression \( \frac{x+2}{x} \) which represents one of the angles. An algebraic expression can often be simplified to make it easier to understand or work with.

Simplification involves combining like terms and performing any possible arithmetic operations. For example, simplifying \( y = 180 - \frac{x+2}{x} \) involves rearranging and simplifying to \( y = 179 - \frac{2}{x} \). This step is crucial as it turns a more complex expression into a friendly form, helping in deeper understanding and easy calculations.

• Simplifying algebraic expressions can also reveal insights about relationships between variables that were not immediately apparent.
• This is particularly useful in geometry where these expressions often tie into key properties of shapes and angles.

Mastering algebraic expressions is vital for solving a wide range of problems in mathematics.