To find the measure of [tex]\( \angle 1 \)[/tex], follow the steps below:
1. Understand the relationship between the angles:
Since [tex]\( \angle 1 \)[/tex] and [tex]\( \angle 2 \)[/tex] form a linear pair, they are supplementary angles. This means their measures add up to 180 degrees.
2. Express the given information in terms of equations:
[tex]\[
m \angle 1 = 6x + 19
\][/tex]
[tex]\[
m \angle 2 = 5x - 4
\][/tex]
3. Set up the equation illustrating their supplementary nature:
[tex]\[
(6x + 19) + (5x - 4) = 180
\][/tex]
4. Combine like terms:
[tex]\[
6x + 19 + 5x - 4 = 180
\][/tex]
[tex]\[
11x + 15 = 180
\][/tex]
5. Solve for x:
[tex]\[
11x + 15 = 180
\][/tex]
Subtract 15 from both sides:
[tex]\[
11x = 165
\][/tex]
Divide by 11:
[tex]\[
x = 15
\][/tex]
6. Find [tex]\( m \angle 1 \)[/tex]:
Substitute [tex]\( x = 15 \)[/tex] back into the expression for [tex]\( m \angle 1 \)[/tex]:
[tex]\[
m \angle 1 = 6x + 19
\][/tex]
[tex]\[
m \angle 1 = 6(15) + 19
\][/tex]
[tex]\[
m \angle 1 = 90 + 19
\][/tex]
[tex]\[
m \angle 1 = 109
\][/tex]
Therefore, the measure of [tex]\( \angle 1 \)[/tex] is [tex]\( 109^\circ \)[/tex].
So, the correct answer is (b) [tex]\( 109^\circ \)[/tex].
