To determine the measure of [tex]\(\angle RST\)[/tex] represented by the expression [tex]\((6x + 12)^{\circ}\)[/tex], we need to evaluate this expression using the given multiple choice options. We will test each option to see which value of [tex]\( x \)[/tex] satisfies the given expression:
1. Option 1: [tex]\( 78^\circ \)[/tex]
We set up the equation:
[tex]\[
6x + 12 = 78
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
6x = 78 - 12 \\
6x = 66 \\
x = \frac{66}{6} \\
x = 11
\][/tex]
This is indeed a valid solution for [tex]\( x \)[/tex].
2. Option 2: [tex]\( 84^\circ \)[/tex]
We set up the equation:
[tex]\[
6x + 12 = 84
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
6x = 84 - 12 \\
6x = 72 \\
x = \frac{72}{6} \\
x = 12
\][/tex]
This is also a valid solution for [tex]\( x \)[/tex].
3. Option 3: [tex]\( 120^\circ \)[/tex]
We set up the equation:
[tex]\[
6x + 12 = 120
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
6x = 120 - 12 \\
6x = 108 \\
x = \frac{108}{6} \\
x = 18
\][/tex]
Again, this is a valid solution for [tex]\( x \)[/tex].
4. Option 4: [tex]\( 156^\circ \)[/tex]
We set up the equation:
[tex]\[
6x + 12 = 156
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
6x = 156 - 12 \\
6x = 144 \\
x = \frac{144}{6} \\
x = 24
\][/tex]
This is also a valid solution for [tex]\( x \)[/tex].
Based on our evaluations above, the valid measure for [tex]\(\angle RST\)[/tex] can be any of the options provided. However, the first valid solution we find is for [tex]\( 78^\circ \)[/tex].
Therefore, [tex]\( m \angle RST = 78^\circ \)[/tex].
