To determine the measure of [tex]\(\angle RST\)[/tex] represented by the expression [tex]\((6x + 12)^\circ\)[/tex], let’s find the appropriate value of [tex]\(x\)[/tex] that results in an angle matching one of the given options. The options are [tex]\(78^\circ\)[/tex], [tex]\(84^\circ\)[/tex], [tex]\(120^\circ\)[/tex], and [tex]\(156^\circ\)[/tex].
Given the expression [tex]\((6x + 12)^\circ\)[/tex], we want to see if substituting different values of [tex]\(x\)[/tex] results in one of these specified angles.
Starting with evaluating different values for [tex]\(x\)[/tex]:
1. If we assume the angle [tex]\(m \angle RST\)[/tex] is [tex]\(84^\circ\)[/tex]:
- We need to solve the equation [tex]\(6x + 12 = 84\)[/tex].
- Subtract 12 from both sides: [tex]\(6x = 84 - 12\)[/tex].
- Simplify: [tex]\(6x = 72\)[/tex].
- Divide by 6: [tex]\(x = 12\)[/tex].
2. Verification:
- Using [tex]\(x = 12\)[/tex]:
[tex]\[
m \angle RST = 6x + 12 = 6(12) + 12 = 72 + 12 = 84^\circ
\][/tex]
This confirms that when [tex]\(x = 12\)[/tex], the measure of [tex]\(\angle RST\)[/tex] is indeed [tex]\(84^\circ\)[/tex], which is one of the given options.
Thus, the measure of [tex]\(m \angle RST\)[/tex] in degrees is [tex]\(\boxed{84^\circ}\)[/tex].
