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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![### Geometry Problem: Calculating Angle \( m \angle ATL \)
**Problem Statement:**
If \( \overline{AB} \perp \overline{MN} \), \( m \angle ATL = (6x - 12)^\circ \) and \( m \angle LTN = (8x - 24)^\circ \), find \( m \angle ATL \).
**Diagram Explanation:**
The diagram accompanying the problem consists of two perpendicular lines intersecting at point \( T \). The lines are labeled as follows:
- \( \overline{AB} \) is a horizontal line.
- \( \overline{MN} \) is a vertical line intersecting \( \overline{AB} \).
- Point \( T \) is where \( \overline{AB} \) and \( \overline{MN} \) intersect.
- Line \( \overline{LT} \) extends diagonally from \( T \), creating angles with \( \overline{AB} \) and \( \overline{MN} \).
The angles associated with this diagram are:
- \( m \angle ATL \)
- \( m \angle LTN \)
**Choices for the Angle \( m \angle ATL \):**
a. \( 48^\circ \)
b. \( 90^\circ \)
c. \( 42^\circ \)
d. \( 76^\circ \)
### Solution Steps:
1. Given \( \overline{AB} \perp \overline{MN} \), this implies that:
\[ m \angle ATN = 90^\circ \]
2. Since \( \angle ATL \) and \( \angle LTN \) together form the right angle \( \angle ATN \):
\[ m \angle ATL + m \angle LTN = 90^\circ \]
3. Substitute the given expressions for the angles:
\[ (6x - 12)^\circ + (8x - 24)^\circ = 90^\circ \]
4. Combine like terms:
\[ 14x - 36 = 90 \]
5. Solve for \( x \):
\[ 14x = 126 \]
\[ x = 9 \]
6. Substitute \( x = 9 \) back into \( m \angle ATL \):
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Transcribed Image Text:### Geometry Problem: Calculating Angle \( m \angle ATL \)
**Problem Statement:**
If \( \overline{AB} \perp \overline{MN} \), \( m \angle ATL = (6x - 12)^\circ \) and \( m \angle LTN = (8x - 24)^\circ \), find \( m \angle ATL \).
**Diagram Explanation:**
The diagram accompanying the problem consists of two perpendicular lines intersecting at point \( T \). The lines are labeled as follows:
- \( \overline{AB} \) is a horizontal line.
- \( \overline{MN} \) is a vertical line intersecting \( \overline{AB} \).
- Point \( T \) is where \( \overline{AB} \) and \( \overline{MN} \) intersect.
- Line \( \overline{LT} \) extends diagonally from \( T \), creating angles with \( \overline{AB} \) and \( \overline{MN} \).
The angles associated with this diagram are:
- \( m \angle ATL \)
- \( m \angle LTN \)
**Choices for the Angle \( m \angle ATL \):**
a. \( 48^\circ \)
b. \( 90^\circ \)
c. \( 42^\circ \)
d. \( 76^\circ \)
### Solution Steps:
1. Given \( \overline{AB} \perp \overline{MN} \), this implies that:
\[ m \angle ATN = 90^\circ \]
2. Since \( \angle ATL \) and \( \angle LTN \) together form the right angle \( \angle ATN \):
\[ m \angle ATL + m \angle LTN = 90^\circ \]
3. Substitute the given expressions for the angles:
\[ (6x - 12)^\circ + (8x - 24)^\circ = 90^\circ \]
4. Combine like terms:
\[ 14x - 36 = 90 \]
5. Solve for \( x \):
\[ 14x = 126 \]
\[ x = 9 \]
6. Substitute \( x = 9 \) back into \( m \angle ATL \):
\
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