Problem 38 - Introductory and Intermediate Algebra for College Students 4th

Question

Find the measure of the complement of each angle. $$41^{\circ}$$

Step-by-Step Solution

Answer

The complement of the angle \(41^{\circ}\) is \(49^{\circ}\).

Step 1: Understanding Complementary Angles

Firstly, understand that complementary angles are two angles whose measures add up to \(90^{\circ}\). So, if you have one angle measure, let's say \(41^{\circ}\), its complement is the measure that you need to add to \(41^{\circ}\) to get \(90^{\circ}\).

Step 2: Finding the Complement

Subtract the given angle from \(90^{\circ}\) to find the complement. In this case, subtract \(41^{\circ}\) from \(90^{\circ}\). This comes to \(90^{\circ} - 41^{\circ} = 49^{\circ}\).

Key Concepts

Understanding Angle Measurement

Angle measurement is a fundamental part of geometry that helps us understand the rotation needed between two intersecting lines. Angles are usually measured in degrees, represented by the symbol \(^{\circ}\).
For instance, a right angle, which is formed by dividing a circle into four equal parts, measures \(90^{\circ}\). Understanding how to measure angles, such as with protractors or through calculations, is crucial for many applications in mathematics.
Key concepts to remember include:

• Zero degrees (\(0^{\circ}\)) indicates no rotation.
• Angles greater than \(0^{\circ}\) and less than \(90^{\circ}\) are called acute angles.
• Angles exactly equal to \(90^{\circ}\) are right angles.
• Angles greater than \(90^{\circ}\) but less than \(180^{\circ}\) are obtuse angles.
• Complementary angles specifically add up to \(90^{\circ}\).

Understanding these basics helps greatly in tackling various geometric problems and is essential when working with complementary angles.

Steps in Problem-Solving

Problem-solving in geometry often involves a step-by-step approach. Start by clearly defining what you are asked to find. In this exercise, the goal is to find the complement of an angle.
Recognize that for two angles to be complementary, their measurements must total \(90^{\circ}\). With the given angle, such as \(41^{\circ}\), use subtraction to find the complement.
Here's how you approach it:

• Identify the given angle (e.g., \(41^{\circ}\)).
• Remember that complementary angles add up to \(90^{\circ}\).
• Subtract the known angle from \(90^{\circ}\): \(90^{\circ} - 41^{\circ} = 49^{\circ}\).

This method exemplifies using basic arithmetic to solve geometric problems efficiently.

Basic Geometry Concepts

In basic geometry, understanding different types of angles and their relationships is essential. Geometry is the branch of mathematics that deals with shapes, sizes, and the properties of space.
Here, angles are a key element, and they can be connected in various ways:

• Complementary Angles: As in this exercise, if two angles add up to \(90^{\circ}\), they are known as complementary.
• Supplementary Angles: These angles add up to \(180^{\circ}\), like a straight line.
• Adjacent Angles: Two angles that share a common side and vertex, lying next to each other.
• Vertical Angles: Angles opposite each other when two lines cross; they are always equal.

Understanding these relationships helps solve problems that involve angle measurements and figure out unknown angles, just like finding the complement in this exercise.