**Problem** In the triangle below, suppose that \( m \angle L = (4x - 3)^\circ \), \( m \angle M = (3x + 7)^\circ \), and \( m \angle N = x^\circ \). Find the degree measure of each angle in the triangle. **Diagram Explanation** The diagram is of a triangle \( \Delta LMN \). The vertices of the triangle are labeled as follows: - Vertex \( L \) with angle measure \( (4x - 3)^\circ \) - Vertex \( M \) with angle measure \( (3x + 7)^\circ \) - Vertex \( N \) with angle measure \( x^\circ \) There is a smaller boxed area to the right of the triangle labeled "Find the degree measure of each angle" with three input boxes next to \( m \angle L \), \( m \angle M \), and \( m \angle N \) for answers in degrees. **Solution** To find the measures of each angle in the triangle, we use the fact that the sum of the angles in a triangle is always \( 180^\circ \). So, we have: \[ m \angle L + m \angle M + m \angle N = 180^\circ \] Substitute the given expressions into this equation: \[ (4x - 3) + (3x + 7) + x = 180 \] Simplify and solve for \( x \): \[ 4x - 3 + 3x + 7 + x = 180 \] Combine like terms: \[ 8x + 4 = 180 \] Subtract 4 from both sides: \[ 8x = 176 \] Divide both sides by 8: \[ x = 22 \] Now, substitute \( x = 22 \) back into the expressions for the measures of the angles: \[ m \angle L = 4x - 3 = 4(22) - 3 = 88 - 3 = 85^\circ \] \[ m \angle M = 3x + 7 = 3(22) + 7 = 66 + 7 = 73^\circ \] \[ m \angle N = x = 22^\circ \] So, the measures of the angles in the triangle are: \[ m \angle L = 85
**Problem** In the triangle below, suppose that \( m \angle L = (4x - 3)^\circ \), \( m \angle M = (3x + 7)^\circ \), and \( m \angle N = x^\circ \). Find the degree measure of each angle in the triangle. **Diagram Explanation** The diagram is of a triangle \( \Delta LMN \). The vertices of the triangle are labeled as follows: - Vertex \( L \) with angle measure \( (4x - 3)^\circ \) - Vertex \( M \) with angle measure \( (3x + 7)^\circ \) - Vertex \( N \) with angle measure \( x^\circ \) There is a smaller boxed area to the right of the triangle labeled "Find the degree measure of each angle" with three input boxes next to \( m \angle L \), \( m \angle M \), and \( m \angle N \) for answers in degrees. **Solution** To find the measures of each angle in the triangle, we use the fact that the sum of the angles in a triangle is always \( 180^\circ \). So, we have: \[ m \angle L + m \angle M + m \angle N = 180^\circ \] Substitute the given expressions into this equation: \[ (4x - 3) + (3x + 7) + x = 180 \] Simplify and solve for \( x \): \[ 4x - 3 + 3x + 7 + x = 180 \] Combine like terms: \[ 8x + 4 = 180 \] Subtract 4 from both sides: \[ 8x = 176 \] Divide both sides by 8: \[ x = 22 \] Now, substitute \( x = 22 \) back into the expressions for the measures of the angles: \[ m \angle L = 4x - 3 = 4(22) - 3 = 88 - 3 = 85^\circ \] \[ m \angle M = 3x + 7 = 3(22) + 7 = 66 + 7 = 73^\circ \] \[ m \angle N = x = 22^\circ \] So, the measures of the angles in the triangle are: \[ m \angle L = 85
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**Problem**
In the triangle below, suppose that \( m \angle L = (4x - 3)^\circ \), \( m \angle M = (3x + 7)^\circ \), and \( m \angle N = x^\circ \).
Find the degree measure of each angle in the triangle.
**Diagram Explanation**
The diagram is of a triangle \( \Delta LMN \). The vertices of the triangle are labeled as follows:
- Vertex \( L \) with angle measure \( (4x - 3)^\circ \)
- Vertex \( M \) with angle measure \( (3x + 7)^\circ \)
- Vertex \( N \) with angle measure \( x^\circ \)
There is a smaller boxed area to the right of the triangle labeled "Find the degree measure of each angle" with three input boxes next to \( m \angle L \), \( m \angle M \), and \( m \angle N \) for answers in degrees.
**Solution**
To find the measures of each angle in the triangle, we use the fact that the sum of the angles in a triangle is always \( 180^\circ \).
So, we have:
\[ m \angle L + m \angle M + m \angle N = 180^\circ \]
Substitute the given expressions into this equation:
\[ (4x - 3) + (3x + 7) + x = 180 \]
Simplify and solve for \( x \):
\[ 4x - 3 + 3x + 7 + x = 180 \]
Combine like terms:
\[ 8x + 4 = 180 \]
Subtract 4 from both sides:
\[ 8x = 176 \]
Divide both sides by 8:
\[ x = 22 \]
Now, substitute \( x = 22 \) back into the expressions for the measures of the angles:
\[ m \angle L = 4x - 3 = 4(22) - 3 = 88 - 3 = 85^\circ \]
\[ m \angle M = 3x + 7 = 3(22) + 7 = 66 + 7 = 73^\circ \]
\[ m \angle N = x = 22^\circ \]
So, the measures of the angles in the triangle are:
\[ m \angle L = 85](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61c70154-6dcc-4ad1-b23f-82f068f0fd01%2F30a070be-d0ad-45c0-b0bc-206cd05c9b4b%2Forl8pr.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem**
In the triangle below, suppose that \( m \angle L = (4x - 3)^\circ \), \( m \angle M = (3x + 7)^\circ \), and \( m \angle N = x^\circ \).
Find the degree measure of each angle in the triangle.
**Diagram Explanation**
The diagram is of a triangle \( \Delta LMN \). The vertices of the triangle are labeled as follows:
- Vertex \( L \) with angle measure \( (4x - 3)^\circ \)
- Vertex \( M \) with angle measure \( (3x + 7)^\circ \)
- Vertex \( N \) with angle measure \( x^\circ \)
There is a smaller boxed area to the right of the triangle labeled "Find the degree measure of each angle" with three input boxes next to \( m \angle L \), \( m \angle M \), and \( m \angle N \) for answers in degrees.
**Solution**
To find the measures of each angle in the triangle, we use the fact that the sum of the angles in a triangle is always \( 180^\circ \).
So, we have:
\[ m \angle L + m \angle M + m \angle N = 180^\circ \]
Substitute the given expressions into this equation:
\[ (4x - 3) + (3x + 7) + x = 180 \]
Simplify and solve for \( x \):
\[ 4x - 3 + 3x + 7 + x = 180 \]
Combine like terms:
\[ 8x + 4 = 180 \]
Subtract 4 from both sides:
\[ 8x = 176 \]
Divide both sides by 8:
\[ x = 22 \]
Now, substitute \( x = 22 \) back into the expressions for the measures of the angles:
\[ m \angle L = 4x - 3 = 4(22) - 3 = 88 - 3 = 85^\circ \]
\[ m \angle M = 3x + 7 = 3(22) + 7 = 66 + 7 = 73^\circ \]
\[ m \angle N = x = 22^\circ \]
So, the measures of the angles in the triangle are:
\[ m \angle L = 85
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