If AB MN, m < ATL = (6x – 12)° and m< LTN = (8x- 24)°, find %3D т<АTL. а. 48° b. 90° С. 42° -> d. 76°

Elementary Geometry For College Students, 7e
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ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
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ChapterP: Preliminary Concepts
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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 \( m \angle ATL \):
   \
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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