Problem 44 - Introductory and Intermediate Algebra for College Students 4th

Question

Find the measure of the supplement of each angle. $$179 .5^{\circ}$$

Step-by-Step Solution

Answer

The measure of the supplement of the angle \(179.5^\circ\) is \(0.5^\circ\).

Step 1: Understand the concept of a supplementary angle

A supplementary angle is the amount of degrees that when added to the given angle equals 180 degrees. We can find this by subtracting the given angle from 180.

Step 2: Subtract the given angle from 180 to find the supplement

So using the concept from step 1, subtract the given angle of \(179.5^\circ\) from 180: \(180^\circ - 179.5^\circ\). After subtracting, you find the supplement of the angle.

Key Concepts

Algebra for College Students

When college students begin to explore algebra, they encounter problems that involve unknown quantities and the use of variables to represent these quantities. One such application is in the domain of geometry, particularly with angle measurements where algebraic expressions and equations are used to find missing angle measures.

In problems like finding the measure of the supplementary angle, algebra serves as a foundation for setting up and solving equations. For instance, suppose you have an angle with a measure of \(x\) degrees. If you are to find its supplement, algebra teaches us to set up an equation like \(x + y = 180\), since the sum of supplementary angles is \(180^\bigcirc\). Here, \(y\) would represent the supplement. Algebra helps translate geometric concepts into equations that can then be manipulated to find the answer, streamlining the process of solving more complex problems involving geometry.

Angle Measurement

Angle measurement is a fundamental concept in geometry that pertains to the magnitude of the angle formed between two rays. Angles are measured in degrees (\(^\bigcirc\)), radians, or gradians, with degrees being the most commonly used unit in most academic settings.

An angle's measurement can tell us a lot about its nature. For example, acute angles are less than \(90^\bigcirc\), right angles are exactly \(90^\bigcirc\), obtuse angles are greater than \(90^\bigcirc\) but less than \(180^\bigcirc\), and straight angles are precisely \(180^\bigcirc\). Understanding how to measure angles accurately is crucial as it forms the basis for many other geometric properties and theorem proofs. Practical applications of angle measurements are seen in fields like engineering, architecture, and even art.

Subtraction of Angles

The subtraction of angles is a simple yet vital operation in geometry that allows us to find the difference between two angle measures. This operation is essential when determining the supplementary angle for a given angle.

The formula for finding supplementary angles is straightforward: the supplement of an angle \(A\) can be found by subtracting its measure from \(180^\bigcirc\), formally written as \(180^\bigcirc - A\).

• When \(A\) is given, one simply subtracts this from \(180^\bigcirc\) to find the angle that, when added to \(A\), equals \(180^\bigcirc\), completing a straight line or a half-circle.
• The concept is based on the geometric property that the sum of angles that form a straight line is exactly \(180^\bigcirc\).

When solving for supplementary angles, precision in calculation is paramount to ensure that both the given angle and its supplement correctly add up to \(180^\bigcirc\), as any deviation will violate basic geometric principles.