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
Related questions
Question
![**Problem Statement:**
Given \( m \parallel n \), find the value of \( x \) and \( y \).
**Diagram Explanation:**
- The diagram shows two parallel lines \( m \) and \( n \) with a transversal crossing them.
- The transversal creates several angles:
- An angle labeled \( y^\circ \) on the line \( m \).
- An angle labeled \( (9x-19)^\circ \) adjacent to \( y^\circ \).
- An angle labeled \( (5x-11)^\circ \) on the line \( n \), corresponding to the alternate interior of the angle by \( y^\circ \).
**Mathematical Setup:**
Using the properties of parallel lines cut by a transversal, we can apply the alternate interior angles theorem, which states that alternate interior angles are equal when two lines are parallel.
\[
y = 5x - 11
\]
And since corresponding angles on parallel lines are equal:
\[
y = 9x - 19
\]
**Equations:**
1. \( y = 5x - 11 \)
2. \( y = 9x - 19 \)
**Solution:**
Set the two expressions for \( y \) equal:
\[
5x - 11 = 9x - 19
\]
Solve for \( x \):
\[
8 = 4x \quad \Rightarrow \quad x = 2
\]
Substitute \( x = 2 \) back into the equation for \( y \):
\[
y = 5(2) - 11 = 10 - 11 = -1
\]
So, the values are:
\[
x = 2, \quad y = -1
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Transcribed Image Text:**Problem Statement:**
Given \( m \parallel n \), find the value of \( x \) and \( y \).
**Diagram Explanation:**
- The diagram shows two parallel lines \( m \) and \( n \) with a transversal crossing them.
- The transversal creates several angles:
- An angle labeled \( y^\circ \) on the line \( m \).
- An angle labeled \( (9x-19)^\circ \) adjacent to \( y^\circ \).
- An angle labeled \( (5x-11)^\circ \) on the line \( n \), corresponding to the alternate interior of the angle by \( y^\circ \).
**Mathematical Setup:**
Using the properties of parallel lines cut by a transversal, we can apply the alternate interior angles theorem, which states that alternate interior angles are equal when two lines are parallel.
\[
y = 5x - 11
\]
And since corresponding angles on parallel lines are equal:
\[
y = 9x - 19
\]
**Equations:**
1. \( y = 5x - 11 \)
2. \( y = 9x - 19 \)
**Solution:**
Set the two expressions for \( y \) equal:
\[
5x - 11 = 9x - 19
\]
Solve for \( x \):
\[
8 = 4x \quad \Rightarrow \quad x = 2
\]
Substitute \( x = 2 \) back into the equation for \( y \):
\[
y = 5(2) - 11 = 10 - 11 = -1
\]
So, the values are:
\[
x = 2, \quad y = -1
\]
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