To determine if lines [tex]\( m \)[/tex] and [tex]\( I \)[/tex] are parallel, we need to check if the given angles [tex]\( m23 \)[/tex] and [tex]\( m27 \)[/tex] are equal.
1. Calculate [tex]\( m23 \)[/tex] when [tex]\( x = 15 \)[/tex]:
Given:
[tex]\[
m23 = (4x + 12)^\circ
\][/tex]
Substitute [tex]\( x = 15 \)[/tex]:
[tex]\[
m23 = 4(15) + 12
\][/tex]
[tex]\[
m23 = 60 + 12
\][/tex]
[tex]\[
m23 = 72^\circ
\][/tex]
2. Calculate [tex]\( m27 \)[/tex] when [tex]\( x = 15 \)[/tex]:
Given:
[tex]\[
m27 = (80 - x)^\circ
\][/tex]
Substitute [tex]\( x = 15 \)[/tex]:
[tex]\[
m27 = 80 - 15
\][/tex]
[tex]\[
m27 = 65^\circ
\][/tex]
3. Compare [tex]\( m23 \)[/tex] and [tex]\( m27 \)[/tex]:
Since:
[tex]\[
m23 = 72^\circ \quad \text{and} \quad m27 = 65^\circ
\][/tex]
[tex]\( m23 \neq m27 \)[/tex], therefore, the lines [tex]\( m \)[/tex] and [tex]\( I \)[/tex] are not parallel when [tex]\( x = 15 \)[/tex].
4. Find the value of [tex]\( x \)[/tex] that makes [tex]\( m23 \)[/tex] and [tex]\( m27 \)[/tex] equal (and hence the lines parallel):
Set [tex]\( m23 \)[/tex] equal to [tex]\( m27 \)[/tex]:
[tex]\[
4x + 12 = 80 - x
\][/tex]
Combine like terms:
[tex]\[
4x + x + 12 = 80
\][/tex]
[tex]\[
5x + 12 = 80
\][/tex]
Isolate [tex]\( x \)[/tex]:
[tex]\[
5x = 80 - 12
\][/tex]
[tex]\[
5x = 68
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{68}{5}
\][/tex]
[tex]\[
x = 13.6
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] that will make lines [tex]\( m \)[/tex] and [tex]\( I \)[/tex] parallel is [tex]\( x = 13.6 \)[/tex].
