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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![### Understanding Parallel Lines and Alternate Interior Angles
**Problem:**
Given \( m \parallel n \), find the value of \( x \).
**Explanation of Diagram:**
The diagram presents two parallel lines \( m \) and \( n \) intersected by a transversal. The angles created at the intersections of the transversal with lines \( m \) and \( n \) are given by algebraic expressions. Specifically:
- The angle on the upper intersection with line \( m \) is labeled as \((5x - 7)^\circ\).
- The angle on the lower intersection with line \( n \) is labeled as \((4x + 6)^\circ\).
**Concept:**
When a transversal intersects two parallel lines, alternate interior angles are equal. That is, these angles are congruent.
**Solution:**
Set the measures of the alternate interior angles equal to each other:
\[ (5x - 7)^\circ = (4x + 6)^\circ \]
Step-by-step solution:
1. **Set up the equation:**
\[ 5x - 7 = 4x + 6 \]
2. **Isolate \( x \):**
\[ 5x - 4x = 6 + 7 \]
\[ x = 13 \]
The value of \( x \) is \( 13 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fddc6b782-5568-4e4b-a41c-c5410dd3aea4%2Fc5f19b00-4060-4838-b888-aa9749df099c%2Fqal2wq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Parallel Lines and Alternate Interior Angles
**Problem:**
Given \( m \parallel n \), find the value of \( x \).
**Explanation of Diagram:**
The diagram presents two parallel lines \( m \) and \( n \) intersected by a transversal. The angles created at the intersections of the transversal with lines \( m \) and \( n \) are given by algebraic expressions. Specifically:
- The angle on the upper intersection with line \( m \) is labeled as \((5x - 7)^\circ\).
- The angle on the lower intersection with line \( n \) is labeled as \((4x + 6)^\circ\).
**Concept:**
When a transversal intersects two parallel lines, alternate interior angles are equal. That is, these angles are congruent.
**Solution:**
Set the measures of the alternate interior angles equal to each other:
\[ (5x - 7)^\circ = (4x + 6)^\circ \]
Step-by-step solution:
1. **Set up the equation:**
\[ 5x - 7 = 4x + 6 \]
2. **Isolate \( x \):**
\[ 5x - 4x = 6 + 7 \]
\[ x = 13 \]
The value of \( x \) is \( 13 \).
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