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Chapter 9: Problem 28

Find the measure of a complement of each angle. $$ 54^{\circ} $$

Short Answer

Expert verified
The complement of \(54^{\text{掳}}\) is \(36^{\text{掳}}\).

Step by step solution

01

Understand complementing angles

Recall that two angles are complementary if the sum of their measures is 90 degrees.
02

Write the equation

Let the measure of the complement of the angle be denoted as \(x\). Thus the equation is: \(54^{\text{掳}} + x = 90^{\text{掳}}\)
03

Solve for \(x\)

Subtract \(54^{\text{掳}}\) from both sides of the equation: \(x = 90^{\text{掳}} - 54^{\text{掳}}\)
04

Simplify the expression

Perform the subtraction: \(x = 36^{\text{掳}}\)

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Key Concepts

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

Basic Geometry
In geometry, we study points, lines, shapes, and angles. It's a field that helps us understand and describe the world around us. Think of geometry as a toolbox filled with different tools that help us solve various problems related to shapes and sizes.
  • Points: They represent a specific location and have no size.
  • Lines: They extend infinitely in both directions and have no width.
  • Shapes: These are formed by connecting points with lines.
  • Angles: These are formed where two lines meet or intersect.
Understanding the basics of geometry is crucial as it forms the foundation for more complex concepts. So, let's dive deeper into the idea of angles and their measurements.
Angle Measurement
Angles are a fundamental part of geometry. They help us describe the space between two intersecting lines. To measure an angle, we use degrees, abbreviated as 掳.
  • A full circle is 360掳.
  • A straight line is 180掳.
  • A right angle is 90掳.
  • An acute angle is less than 90掳.
  • An obtuse angle is more than 90掳 but less than 180掳.
By knowing these basic angle measurements, we can describe and solve many geometric problems, like finding complementary angles.
Complementary Angles
Complementary angles are two angles whose measures add up to 90 degrees. It's like they complete each other to form a right angle. For instance, if you have one angle that measures 54掳, you need to find another angle that, when added to 54掳, equals 90掳.
Mathematically, it can be expressed as:
\( x + 54^{\circ} = 90^{\circ} \)
Solving this, you get:
\( x = 90^{\circ} - 54^{\circ} \)
This means \( x = 36^{\circ} \).
Therefore, the complementary angle of 54掳 is 36掳.
Problem-Solving Steps
When solving problems related to complementary angles, follow these steps:
  • Step 1: Understand the definition of complementary angles. They add up to 90掳.
  • Step 2: Write down the equation that represents the problem. For example, if you need to find the complement of 54掳, write down \( 54^{\circ} + x = 90^{\circ} \).
  • Step 3: Solve for the unknown angle, x. In this case, subtract 54掳 from both sides of the equation: \( x = 90^{\circ} - 54^{\circ} \).
  • Step 4: Simplify the expression. Here, you will find that \( x = 36^{\circ} \).
It's important to practice these steps with different angles so you get comfortable with the concept of complementary angles and how to find them.

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