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Chapter 7: Problem 68

Find the measure of an angle whose supplement is three times its measure.

Short Answer

Expert verified
The measure of the angle is 45 degrees.

Step by step solution

01

Understand the Relationship

A supplementary angle means that the two angles add up to 180 degrees. Let's denote the angle we need to find as \(x\). Its supplement would therefore be \(180 - x\).
02

Set Up the Equation

The problem states that the supplement of the angle is three times the measure of the angle. This can be written as the equation: \[ 180 - x = 3x \]
03

Solve for \( x \)

First, add \( x \) to both sides of the equation to get all the terms involving \( x \) on one side: \[ 180 = 4x \] Then, divide both sides by 4 to solve for \( x \): \[ x = \frac{180}{4} = 45 \]

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Key Concepts

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

angle measurement
In geometry, angle measurement is key to understanding relationships between different parts of a figure. Angles are measured in degrees (掳). Supplementary angles are one such relationship where two angles add up to 180 degrees. For instance, if one angle measures 60 degrees, its supplementary angle must be 120 degrees because 60 + 120 = 180. Understanding this concept is critical for solving problems involving angle relationships, like the exercise we are discussing.
equation setup
Setting up the equation correctly is essential to solving geometry problems. In this exercise, we need to find an angle whose supplement is three times its measure. First, we express the unknown angle as x. According to the supplementary angle rule, its supplement will be 180 - x. The problem states that this supplement equals three times the original angle. So, we set up the equation: \[ 180 - x = 3x \]By creating this equation, we effectively translate the problem's words into a mathematical expression we can solve.
algebraic manipulation
Once the equation is set up, the next step is to solve it through algebraic manipulation. Here鈥檚 how to solve our equation step-by-step:Firstly, we start with: \[ 180 - x = 3x \] Then, we add x to both sides to get all the x terms on one side of the equation: \[ 180 = 4x \] Finally, we divide both sides by 4 to isolate x: \[ x = \frac{180}{4} = 45 \]So, the measure of the angle is 45 degrees. Always check your work by substituting x back into the context of the problem to ensure it satisfies the original conditions. In this case, the supplement would be 180 - 45 = 135, and 135 is indeed three times 45, confirming that our solution is correct.

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