Problem 34 - Precalculus: A Concise Course

Question

Find (if possible) the complement and supplement of each angle. (a) 3 (b) 1.5

Step-by-Step Solution

Answer

The complement and supplement of the first angle (3°) are 87° and 177° respectively. The complement and supplement of the second angle (1.5°) are 88.5° and 178.5° respectively.

Step 1: Find complement and supplement of the first angle

First, calculate the complement and supplement of the first angle which is 3 degrees. The complement is calculated by subtracting the angle from 90 degrees and the supplement is calculated by subtracting the angle from 180 degrees. In this case, the complement of 3 degrees will be \(90 - 3 = 87\) degrees and the supplement will be \(180 - 3 = 177\) degrees.

Step 2: Find complement and supplement of the second angle

Secondly, calculate the complement and supplement of the second angle which is 1.5 degrees. As before, the complement is \(90 - 1.5 = 88.5\) degrees and the supplement is \(180 - 1.5 = 178.5\) degrees. This completes the exercise.

Key Concepts

Angle Measurement

Angle measurement is a basic concept in geometry where we determine the size of an angle in degrees. Angles are an important part of mathematics and everyday life, appearing in everything from architecture to navigation.
Angles are commonly measured in degrees, with a full rotation being 360 degrees. Here's a quick overview of some key points about angles:

• An angle is formed by two rays (or line segments) that have a common endpoint, known as the vertex.
• A right angle measures exactly 90 degrees.
• An acute angle is less than 90 degrees, while an obtuse angle is more than 90 but less than 180 degrees.
• A straight angle is precisely 180 degrees, forming a straight line.

Understanding how to measure and calculate angles is crucial for solving many problems in geometry, trigonometry, and beyond.

Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. This means if you know one angle, you can easily determine its complement by subtracting the angle from 90.
For example, if one angle is 3 degrees, its complementary angle is calculated as follows:

• Complement angle = 90 - 3 = 87 degrees.

A practical application of complementary angles is in right-angle triangles, where the two non-right angles are always complementary.
So, whenever you work with complementary angles, just remember that they should add up to form a right angle. Knowing this can help simplify quite a few geometry problems and proofs by providing quick checks for accuracy.

Supplementary Angles

Supplementary angles are two angles whose measures sum to 180 degrees, forming a straight line when placed together. If you know one of the angles, you can find its supplement by subtracting the angle from 180.
For instance, if an angle is 1.5 degrees, the supplementary angle will be:

• Supplementary angle = 180 - 1.5 = 178.5 degrees.

Supplementary angles frequently appear in many geometrical contexts, particularly involving linear pairs where two adjacent angles form a straight line.
Working with supplementary angles helps in checking the consistency and correctness of solutions involving straight angles, as well as providing insights into the spatial arrangement of lines and angles in various geometric settings.