Chapter 2: Problem 67

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

### Short Answer

## Step by step solution

## Identify the given variables

## Express the supplement in terms of \( x \)

## Set up the equation

## Solve the equation

## Verify the solution

## Key Concepts

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

###### Understanding Algebraic Equations

- We first identify the given variables.
- Next, we express all quantities in terms of \( x \).
- Finally, we set up and solve the equation for \( x \).

###### Angle Measurements

A complete circle measures \( 360^\text{°} \), and a straight line measures \( 180^\text{°} \). The exercise involves determining the value of an unknown angle whose supplement is given in a specific ratio.

Understanding how to measure angles and calculate their supplements is crucial for solving complex geometrical problems. Remember:

- A right angle equals \( 90^\text{°} \).
- An acute angle is less than \( 90^\text{°} \).
- An obtuse angle is greater than \( 90^\text{°} \) but less than \( 180^\text{°} \).

###### Angle Relationships

The key types of relationships between angles include:

- Complementary Angles: Two angles whose measures sum up to \( 90^\text{°} \). For instance, \( 30^\text{°} \) and \( 60^\text{°} \).
- Supplementary Angles: Two angles whose measures sum up to \( 180^\text{°} \). For example, \( 110^\text{°} \) and \( 70^\text{°} \).
- Adjacent Angles: Two angles that share a common side and vertex (corner point).

###### Supplementary Angles

For example, if one angle is \( x \), its supplement is \( 180^\text{°} - x \). This relationship helps us set up equations to solve problems involving supplementary angles.

In the given exercise, we denote the unknown angle as \( x \). Therefore, its supplement can be written as \( 180^\text{°} - x \). Since the problem states the supplement is eight times the original angle, we set up the equation \( 180^\text{°} - x = 8x \). By solving this, we find \( x \). Understanding this concept helps solve not just this problem, but also many other geometry-related problems.