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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![## Transversal Problem
### Diagram Description:
The diagram above consists of two parallel lines, \( \overleftrightarrow{AB} \) and \( \overleftrightarrow{CD} \), intersected by the transversal \( \overline{KL} \). Two angles are noted: \(\angle LRB\) and \(\angle RSC\), which appear to be consecutive interior angles on the same side of the transversal.
### Given:
\[
\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}
\]
\[
m\angle LRB = (7x + 10)
\]
\[
m\angle RSC = (8x + 5)
\]
### Problem:
Find \( m\angle ARL \) in degrees.
### Solution Steps:
Using the property that consecutive interior angles on parallel lines are supplementary, we can set up the following equation:
\[
m\angle LRB + m\angle RSC = 180^\circ
\]
Substitute the given expressions for the angles:
\[
(7x + 10) + (8x + 5) = 180
\]
Combine like terms:
\[
15x + 15 = 180
\]
Subtract 15 from both sides:
\[
15x = 165
\]
Divide by 15:
\[
x = 11
\]
Now, substitute \( x = 11 \) back into either of the original angle expressions to find \( m\angle ARL \):
\[
m\angle LRB = 7(11) + 10 = 87^\circ
\]
\[
m\angle RSC = 8(11) + 5 = 93^\circ
\]
Since \( \overleftrightarrow{AB} \parallel \overleftrightarrow{CD} \) and \( \angle ARL \) corresponds to \( \angle LRB \) (either alternate exterior or corresponding depending on exact placement),
Therefore, \( m\angle ARL = 87^\circ \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2a5bd6d-382a-4670-905a-c7c57e5f0e16%2F54ebf1be-2943-4a9d-a8f8-a9469d6b0d3d%2Fyenvkeb_processed.png&w=3840&q=75)
Transcribed Image Text:## Transversal Problem
### Diagram Description:
The diagram above consists of two parallel lines, \( \overleftrightarrow{AB} \) and \( \overleftrightarrow{CD} \), intersected by the transversal \( \overline{KL} \). Two angles are noted: \(\angle LRB\) and \(\angle RSC\), which appear to be consecutive interior angles on the same side of the transversal.
### Given:
\[
\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}
\]
\[
m\angle LRB = (7x + 10)
\]
\[
m\angle RSC = (8x + 5)
\]
### Problem:
Find \( m\angle ARL \) in degrees.
### Solution Steps:
Using the property that consecutive interior angles on parallel lines are supplementary, we can set up the following equation:
\[
m\angle LRB + m\angle RSC = 180^\circ
\]
Substitute the given expressions for the angles:
\[
(7x + 10) + (8x + 5) = 180
\]
Combine like terms:
\[
15x + 15 = 180
\]
Subtract 15 from both sides:
\[
15x = 165
\]
Divide by 15:
\[
x = 11
\]
Now, substitute \( x = 11 \) back into either of the original angle expressions to find \( m\angle ARL \):
\[
m\angle LRB = 7(11) + 10 = 87^\circ
\]
\[
m\angle RSC = 8(11) + 5 = 93^\circ
\]
Since \( \overleftrightarrow{AB} \parallel \overleftrightarrow{CD} \) and \( \angle ARL \) corresponds to \( \angle LRB \) (either alternate exterior or corresponding depending on exact placement),
Therefore, \( m\angle ARL = 87^\circ \).
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