Problem 41 - Introductory and Intermediate Algebra for College Students 4th

Question

Find the measure of the supplement of each angle. $$132^{\circ}$$

Step-by-Step Solution

Answer

The supplement of an angle of 132 degrees is 48 degrees.

Step 1 - Understand what a supplementary angle means

Supplementary angles are pairs of angles that add up to 180 degrees. If we are given one angle, we can find its supplement by subtracting that angle from 180 degrees.

Step 2 - Subtract the given angle from 180 degrees

To find the supplement of 132 degrees, subtract 132 from 180: \( 180 - 132 = 48 \) degrees

Key Concepts

Algebra for College Students

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols; it is a language that bridges gaps in higher-level math, such as calculus and linear algebra. When tackling algebraic problems, students must be familiar with operations on algebraic expressions and the use of equations.

In the context of finding supplementary angles, algebraic principles are applied by creating and solving a simple linear equation. For example, if we know one angle measure and need to find its supplement, we would set up an equation like this: If one angle is denoted by the variable 'x', the supplementary angle will be represented as '180 - x'. When given that 'x' equals 132 degrees, we simply compute '180 - x' to get the measure of the supplementary angle.

This process of manipulating equations is essential not just in geometry, but across all fields of algebra, paving the way for solving more complex problems in college-level math courses.

Measuring Angles

Angles are a fundamental component of geometry, representing the space between two intersecting lines or rays. The unit of measure commonly used for angles is degrees, and being adept at measuring angles is crucial for students studying geometry.

To measure an angle, one can use a protractor or apply mathematical principles when the angle size is given. For instance, if a textbook exercise asks to find the measure of an angle's supplement, we use the fact that the total of the measures of supplementary angles is always 180 degrees. Thus, we would subtract the given angle from 180 degrees, as shown in the exercise involving the 132-degree angle.

Importance of Accurate Measurement

Ensuring accurate measurements when dealing with angles is imperative to solving a multitude of geometrical problems, from simple exercises to more complex real-world applications like construction and engineering.

Angle Relationships

Understanding the relationship between angles is an essential skill in geometry. There are various types of angle relationships, such as complementary (summing to 90 degrees), supplementary (summing to 180 degrees), and vertical angles (being equal), among others.

In this exercise, we are focused on supplementary angles. If two angles are supplementary, the sum of their measures is 180 degrees. Knowing this, we can easily find the measure of an unknown angle if its supplement is given. This concept is not only crucial for solving problems in geometry but also helps in spotting patterns and understanding the properties of shapes and designs.

Real-World Applications

Beyond academics, comprehension of angle relationships aids in fields that require spatial reasoning, such as architecture, art, and even aviation. For students, mastering these concepts solidifies foundational geometry knowledge and enhances their problem-solving skillset.