To find the measure of [tex]\( \angle RST \)[/tex], we need to determine the value of [tex]\( x \)[/tex] that makes the expression [tex]\( (6x + 12)^\circ \)[/tex] match one of the provided degree options: [tex]\( 78^\circ \)[/tex], [tex]\( 84^\circ \)[/tex], [tex]\( 120^\circ \)[/tex], or [tex]\( 156^\circ \)[/tex].
Let's check each option one by one:
1. Checking [tex]\(78^\circ\)[/tex]:
- Set the expression equal to 78:
[tex]\[
6x + 12 = 78
\][/tex]
- Subtract 12 from both sides:
[tex]\[
6x = 66
\][/tex]
- Divide by 6:
[tex]\[
x = 11
\][/tex]
- Since [tex]\( x \)[/tex] is an integer, [tex]\( 78^\circ \)[/tex] is a valid solution.
2. Checking [tex]\(84^\circ\)[/tex]:
- Set the expression equal to 84:
[tex]\[
6x + 12 = 84
\][/tex]
- Subtract 12 from both sides:
[tex]\[
6x = 72
\][/tex]
- Divide by 6:
[tex]\[
x = 12
\][/tex]
- Although [tex]\( x \)[/tex] is an integer, we already found a valid solution with [tex]\( x = 11 \)[/tex].
3. Checking [tex]\(120^\circ\)[/tex]:
- Set the expression equal to 120:
[tex]\[
6x + 12 = 120
\][/tex]
- Subtract 12 from both sides:
[tex]\[
6x = 108
\][/tex]
- Divide by 6:
[tex]\[
x = 18
\][/tex]
- Similar to the previous solutions, [tex]\( x = 18 \)[/tex] is also an integer, but we consider the first valid instance acceptable.
4. Checking [tex]\(156^\circ\)[/tex]:
- Set the expression equal to 156:
[tex]\[
6x + 12 = 156
\][/tex]
- Subtract 12 from both sides:
[tex]\[
6x = 144
\][/tex]
- Divide by 6:
[tex]\[
x = 24
\][/tex]
- As with the previous solutions, [tex]\( x = 24 \)[/tex] is an integer, but again, the earliest valid solution is sufficient.
Therefore, the measure of [tex]\( \angle RST \)[/tex] is:
[tex]\[
\boxed{78^\circ}
\][/tex]
