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Chapter 7: Problem 65

Find the measure of an angle whose complement is four times its measure. (Hint: If \(x\) represents the measure of the unknown angle, how would we represent its complement?)

Short Answer

Expert verified
The measure of the angle is 18 degrees.

Step by step solution

01

- Understand the Concept of Complementary Angles

Complementary angles sum up to 90 degrees. If one angle is represented by \(x\), its complement will be \(90 - x\).
02

- Set Up the Equation

According to the problem, the complement of the angle is four times the measure of the angle itself. This gives us the equation: \(90 - x = 4x\).
03

- Solve the Equation

We now solve for \(x\): \[ 90 - x = 4x \] Add \(x\) to both sides: \[ 90 = 5x \] Divide by 5: \[ x = 18 \]
04

- Verify the Solution

The measure of the angle is 18 degrees. The complement of 18 degrees is \(90 - 18 = 72\) degrees. Checking the problem condition: Is 72 four times 18? Yes, \(72 = 4 \times 18\).

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Key Concepts

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

Angle Measurement
Angles are an essential part of geometry and are measured in degrees. There are different types of angles, including complementary angles. Complementary angles are two angles whose measures add up to 90 degrees. For example, if one angle is 60 degrees, its complement would be 30 degrees, because 60 + 30 = 90. Always remember that when dealing with complementary angles, their measures must sum up to exactly 90 degrees.
In this exercise, if given an angle measure of \( x \), we can denote its complement as \( 90 - x \). Understanding this basic concept is vital for solving problems involving complementary angles.
Solving Equations
Solving equations is crucial in mathematics, particularly when solving for unknown values. Let's break down the process for the given problem.
We are told that the complement of an angle is four times the angle itself. If we let \( x \) represent the unknown angle, its complement is \( 90 - x \). According to the problem, this is equal to four times the angle, giving us the equation:
\( 90 - x = 4x \)
To solve for \( x \), follow these steps:
  • Add \( x \) to both sides to get all \( x \) terms on one side: \( 90 = 5x \)
  • Divide both sides by 5 to isolate \( x \): \( x = 18 \)

This solution of \( x = 18 \) means the unknown angle is 18 degrees. Solving equations accurately is the key to finding the correct measure.
Verification of Solutions
After solving an equation, it's always important to verify your solution to ensure it meets the problem's conditions. Verification provides confirmation that your steps are correct and that your answer is accurate.
For this exercise, we found the angle to be 18 degrees. To verify:
  • Calculate the complement of 18 degrees, which is \( 90 - 18 = 72 \) degrees.
  • Check the problem's condition: Is 72 degrees four times 18 degrees? Yes, because \( 72 = 4 \times 18 \).

Since both checks are correct, our solution that the angle measures 18 degrees is verified. Always ensure your solution makes sense and satisfies the given conditions in the problem.

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