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Chapter 4: Problem 3

Find \((a)\) the complement and \((b)\) the supplement of the given angles. $$42^{\circ}$$

Short Answer

Expert verified
The complement is 48掳 and the supplement is 138掳.

Step by step solution

01

Understand Complements

The complement of an angle is another angle that adds up to 90掳 with it. Thus, to find the complement of a given angle, subtract the angle from 90掳.
02

Calculate Complement

Given the angle is 42掳, find the complement.\[90^{\circ} - 42^{\circ} = 48^{\circ}\]
03

Understand Supplements

The supplement of an angle is another angle that adds up to 180掳 with it. Thus, to find the supplement of a given angle, subtract the angle from 180掳.
04

Calculate Supplement

Given the angle is 42掳, find the supplement.\[180^{\circ} - 42^{\circ} = 138^{\circ}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complementary Angles
Complementary angles are two angles whose measures add up to exactly 90 degrees. These angles might be next to each other, forming a right angle, or they might be apart. When dealing with any angles, if you鈥檙e asked for the complementary angle, you find it by subtracting the given angle from 90 degrees.
For example, if your angle is 42 degrees like in the exercise, you just calculate the complement by \[ 90^{\circ} - 42^{\circ} = 48^{\circ} \].
This means that 42 degrees and 48 degrees are complementary because together, they form a right angle.
Understanding complementary angles is essential in various applications such as in trigonometry and geometry where right angles are fundamental.
Supplementary Angles
Supplementary angles are two angles whose measures add up to 180 degrees. These angles often form a straight line when placed adjacent to each other. Knowing how to calculate supplementary angles is useful in geometry, especially when analyzing shapes like triangles and quadrilaterals.
To find the supplementary angle, subtract the given angle from 180 degrees. If you have an angle measuring 42 degrees, the supplement would be \[ 180^{\circ} - 42^{\circ} = 138^{\circ} \].
Therefore, 42 degrees and 138 degrees together make a straight angle, illustrating the nature of supplementary angles. Recognizing these relationships helps in understanding and solving problems involving linear pairs and intersecting lines.
Degree Measure
The degree is a unit of measure for angles. It provides a numerical way to express the size of an angle. We often represent angles with a small circle (掳) symbol after the number, like 42掳, to denote degrees.
This unit is part of a circle, where a full circle is 360 degrees, a right angle is 90 degrees, and a straight angle is 180 degrees. Understanding degree measure is crucial because it helps in communicating and calculating precise angle relationships.
In mathematics, various tools like protractors are used to measure and construct angles accurately in degrees. By knowing the degree measures, students can easily perform operations like finding complementary and supplementary angles, essential for more complex geometric problems.

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Most popular questions from this chapter

Find the smallest positive measure of \(\theta\) (rounded to the nearest degree) if the indicated information is true. \(\sec \theta=-3.1421\) and the terminal side of \(\theta\) lies in quadrant II.

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