Problem 71 - Prealgebra

Question

Find the complement and supplement of each angle. [Example \(6]\) $$31^{\circ}$$

Step-by-Step Solution

Answer

The complement is \(59^{\circ}\) and the supplement is \(149^{\circ}\).

Step 1: Understanding the Problem

We need to find two things for the given angle of \(31^{\circ}\): its complement and its supplement. Let's recall what each of these terms means. The complement of an angle \(x\) is defined as the angle that, when added to \(x\), results in \(90^{\circ}\). The supplement of an angle \(x\) is the angle that, when added to \(x\), equals \(180^{\circ}\).

Step 2: Calculate the Complement

To find the complement of \(31^{\circ}\), we will subtract it from \(90^{\circ}\). Thus, the complement is: \[90^{\circ} - 31^{\circ} = 59^{\circ}.\]

Step 3: Calculate the Supplement

To find the supplement of \(31^{\circ}\), we subtract it from \(180^{\circ}\). Thus, the supplement is: \[180^{\circ} - 31^{\circ} = 149^{\circ}.\]

Key Concepts

Angles

Angles are a fundamental part of geometry. They measure the rotation between two lines that meet at a point, called the vertex. There are different types of angles based on their measure: acute (less than \(90^{\circ}\)), right (exactly \(90^{\circ}\)), obtuse (more than \(90^{\circ} \) but less than \(180^{\circ}\)), straight (exactly \(180^{\circ}\)), and reflex (more than \(180^{\circ}\)). When dealing with angles, it's essential to understand how they relate to each other.

• Complementary angles are two angles that sum up to \(90^{\circ}\). For example, if one angle is \(30^{\circ}\), the other must be \(60^{\circ}\) to be complementary.
• Supplementary angles add up to \(180^{\circ}\). If one angle is \(110^{\circ}\), the supplementary angle would be \(70^{\circ}\).

Recognizing these relationships is crucial in solving angle problems.

Basic Geometry

Basic geometry is the branch of mathematics dealing with shapes, sizes, and properties of space. Understanding angles is vital in geometry because they are used in the formation of polygons and in tessellations. By knowing how to find complementary and supplementary angles, we can better design and analyze geometric figures.

• For triangles, it's important to note that the sum of the interior angles always equals \(180^{\circ}\).
• In quadrilaterals, the sum of interior angles is \(360^{\circ}\), which is a product of having two triangular components.

Mastering these basics helps in advanced explorations of geometry topics.

Mathematical Problem-Solving

Mathematical problem-solving involves applying knowledge and reasoning to find solutions. The process generally includes understanding the problem, devising a plan, carrying out the plan, and looking back to check the work.

• In the case of finding complementary and supplementary angles, understanding the definition and application of these terms is the first step.
• Next, performing calculations using basic subtraction helps to find the required angle, as shown in the solution steps.

Verifying the calculated angles by checking if they indeed sum to \(90^{\circ}\) for complementary, or \(180^{\circ}\) for supplementary, ensures accuracy. This methodology in learning enhances overall critical thinking and problem-solving skills.