Question

the measure of \\( \angle rst \\) can be represented by the expression \\( (6x + 12) ^ { \circ } \\). what is \\( m \angle rst \\) in degrees? \\( 78 ^ { \circ } \\) \\( 84 ^ { \circ } \\) \\( 120 ^ { \circ } \\) \\( 156 ^ { \circ } \\)

Explanation:

Step1: Set up the equation

Since \( \angle RSU = 78^{\circ}\) and \( \angle TSU=(3x - 12)^{\circ}\), and \( \angle RST=(6x + 12)^{\circ}\), and \( \angle RST=\angle RSU+\angle TSU\).
So \(6x + 12=78+(3x - 12)\).

Step2: Solve for \(x\)

Simplify the right - hand side: \(78+(3x - 12)=3x+66\).
The equation becomes \(6x + 12=3x + 66\).
Subtract \(3x\) from both sides: \(6x-3x + 12=3x-3x + 66\), \(3x+12 = 66\).
Subtract \(12\) from both sides: \(3x+12-12=66 - 12\), \(3x=54\).
Divide both sides by \(3\): \(x = 18\).

Step3: Find \(m\angle RST\)

Substitute \(x = 18\) into \(6x + 12\).
\(6\times18+12=108 + 12=120\).

Answer:

\(120^{\circ}\)