Problem 54 - Prealgebra

Question

Two angles are supplementary angles. If one of the angles is \(23^{\circ},\) then solving the equation \(x+23^{\circ}=180^{\circ}\) will give you the other angle. Solve the equation.

Step-by-Step Solution

Answer

The other angle is \(157^{\circ}\).

Step 1: Understand the Definition of Supplementary Angles

Supplementary angles are two angles whose measures add up to \(180^{\circ}\). Since one angle is given as \(23^{\circ}\), the sum of this angle and the unknown angle must equal \(180^{\circ}\).

Step 2: Write Down the Equation

The problem gives us the equation \(x + 23^{\circ} = 180^{\circ}\), where \(x\) represents the measure of the unknown angle.

Step 3: Solve the Equation for x

To find \(x\), subtract \(23^{\circ}\) from both sides of the equation: \[ x = 180^{\circ} - 23^{\circ} \]

Step 4: Perform the Calculation

Calculate \(180^{\circ} - 23^{\circ}\): \(180 - 23 = 157\). Thus, \(x = 157^{\circ}\).

Key Concepts

Equation Solving

Equation solving is a key skill in algebra and geometry. It involves finding the value of an unknown variable that makes an equation true. In the context of supplementary angles, this skill is particularly useful for determining the measure of one angle when the other is known.
To solve an equation, the goal is to isolate the variable on one side. In our example, the equation is given as:

• \(x + 23^{\circ} = 180^{\circ}\)

To isolate \(x\), we perform operations that simplify the equation. By subtracting \(23^{\circ}\) from both sides, we find:

• \(x = 180^{\circ} - 23^{\circ}\)

This operation leaves \(x\) by itself on the left side, showing precisely how these mathematical operations help discover the unknown angle. It’s important to always perform the same operation on both sides of the equation to maintain its balance.

Angle Measurement

Understanding angle measurement is essential in geometry. Angles are measured in degrees, where a full circle is \(360^{\circ}\). Supplementary angles, like in our example, are two angles that add up to \(180^{\circ}\).
When given one angle measure, it's crucial to know how to calculate the other. This example introduces the complementary nature of angles. If the angle provided is \(23^{\circ}\), the simplest way to comprehend the remaining angle is to acknowledge that it is the difference between \(180^{\circ}\) and the given angle. This reflective understanding aids quick computations and verifications in future problems.
Moreover, knowing the ranges and common pairings of angles, such as what makes a right angle or a straight line, aids overall understanding in both theoretical and practical settings.

Subtraction Operation

Subtraction is one of the fundamental arithmetic operations. It involves finding the difference between two numbers. In our equation for supplementary angles, subtraction helps isolate the unknown variable.
Here’s how subtraction is used:

• Start with the equation \(x + 23^{\circ} = 180^{\circ}\).
• Subtract \(23^{\circ}\) from both sides:
  • \(180^{\circ} - 23^{\circ} = 157^{\circ}\).

The subtraction operation is useful not only for solving equations but also for understanding how parts of a whole relate to each other. Check again that your subtraction is correct. Subtraction shows us how much needs to be added to reach a total, which is exactly how we find the missing piece in supplementary angles. Understanding subtraction thoroughly allows you to tackle more complex mathematical problems with confidence.