Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
Set up an equation and solve for each variable. Identify the definition(s) and/or theroem(s) used.
![### Solving for Unknown Angles in Parallel Line Equations
In this diagram, we have two parallel lines that are intersected by a transversal. The alternate interior angles formed by the intersection are expressed algebraically as follows:
- **Angle 1**: \(17x - 4\)
- **Angle 2**: \(12 + 15x\)
According to the properties of parallel lines and a transversal, alternate interior angles are congruent. This means that:
\[ 17x - 4 = 12 + 15x \]
To find the value of \(x\), follow these algebraic steps:
1. **Isolate the variable \(x\)**:
\[
17x - 4 = 12 + 15x
\]
Subtract \(15x\) from both sides:
\[
17x - 15x - 4 = 12
\]
Simplify:
\[
2x - 4 = 12
\]
2. **Solve for \(x\)**:
Add 4 to both sides:
\[
2x = 16
\]
Divide by 2:
\[
x = 8
\]
### Conclusion
By solving the equation \(17x - 4 = 12 + 15x\), we determine that \(x = 8\). This value can then be used to find the specific measures of the angles in the diagram.
This example not only illustrates the principles of algebraic manipulation but also emphasizes the geometric property that alternate interior angles formed by a transversal with parallel lines are equal.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F512160d8-b10f-4611-8441-57db0606efae%2Fe981cfe7-a99a-47ad-8622-80836894a2dd%2Fllcp50d_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving for Unknown Angles in Parallel Line Equations
In this diagram, we have two parallel lines that are intersected by a transversal. The alternate interior angles formed by the intersection are expressed algebraically as follows:
- **Angle 1**: \(17x - 4\)
- **Angle 2**: \(12 + 15x\)
According to the properties of parallel lines and a transversal, alternate interior angles are congruent. This means that:
\[ 17x - 4 = 12 + 15x \]
To find the value of \(x\), follow these algebraic steps:
1. **Isolate the variable \(x\)**:
\[
17x - 4 = 12 + 15x
\]
Subtract \(15x\) from both sides:
\[
17x - 15x - 4 = 12
\]
Simplify:
\[
2x - 4 = 12
\]
2. **Solve for \(x\)**:
Add 4 to both sides:
\[
2x = 16
\]
Divide by 2:
\[
x = 8
\]
### Conclusion
By solving the equation \(17x - 4 = 12 + 15x\), we determine that \(x = 8\). This value can then be used to find the specific measures of the angles in the diagram.
This example not only illustrates the principles of algebraic manipulation but also emphasizes the geometric property that alternate interior angles formed by a transversal with parallel lines are equal.
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