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Chapter 9: Problem 31

Find the measure of a supplement of each angle. $$ 139^{\circ} $$

Short Answer

Expert verified
The supplement of \(139^{\circ}\) is \(41^{\circ}\).

Step by step solution

01

Understand the concept of supplementary angles

Supplementary angles are two angles whose measures add up to 180 degrees. Therefore, if you have one angle, the supplement of this angle is the difference between 180 degrees and the given angle.
02

Set up the equation

Let the given angle be represented by \(A\). The measure of the supplement of angle \(A\) can be found using the equation: \[ \text{Supp}(A) = 180^{\circ} - A \]
03

Substitute the given angle

Substitute \( 139^{\circ} \) for \( A \) in the equation: \[ \text{Supp}(139^{\circ}) = 180^{\circ} - 139^{\circ} \]
04

Calculate the supplement

Perform the subtraction to find the measure of the supplementary angle: \[ \text{Supp}(139^{\circ}) = 180^{\circ} - 139^{\circ} = 41^{\circ} \]

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Key Concepts

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

Angle Measurement
Angle measurement is a fundamental concept in geometry. An angle is formed by two rays (the sides of the angle) that share a common endpoint known as the vertex. Angles are measured in degrees, with a full circle comprising 360 degrees.
To measure an angle, you often use a protractor. The angle is the amount of rotation needed to bring one side to the other. For example:
  • A right angle measures 90 degrees.
  • An acute angle measures less than 90 degrees.
  • An obtuse angle measures more than 90 degrees but less than 180 degrees.
These measurements help us categorize and compare angles in geometric problems, such as finding the supplementary angle.
Supplement of an Angle
The supplement of an angle is a concept in angle relationships. When two angles are supplementary, their measures add up to 180 degrees. This idea is helpful when you know one angle and need to find its supplement.
For instance, if you have angle A, the supplement of A is found using the formula: \( \text{Supp}(A) = 180^{\circ} - A \).
Let's use the example from the exercise:
  • Given angle = 139 degrees
  • Supplement = 180 degrees - 139 degrees = 41 degrees
Therefore, the supplement of a 139-degree angle is a 41-degree angle. This is crucial when solving geometric problems that involve supplementary angles.
Basic Arithmetic Operations
Basic arithmetic operations are essential skills in mathematics and geometry. These operations include addition, subtraction, multiplication, and division. For angle calculations, subtraction is often used to find supplementary angles.
In the context of the exercise, consider the subtraction operation: \( 180^{\circ} - 139^{\circ} \). Here's how you perform this:
  • Start with 180 degrees.
  • Subtract the given angle (139 degrees) from it.
  • Result: 180 - 139 = 41 degrees.
This straightforward subtraction gives you the measure of the supplementary angle. Mastering these basic operations ensures you can tackle more complex problems with confidence.

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