Question

the measure of $\angle rst$ can be represented by the expression $(6x + 12)^circ$. what is $m\angle rst$ in degrees? $\bigcirc\\ 78^circ$ $\bigcirc\\ 84^circ$ $\bigcirc\\ 120^circ$ $\bigcirc\\ 156^circ$ (there is a diagram with points r, s, u, t, angles at s: 78°, $(3x - 12)^circ$, and $\angle rst$ is composed of these angles.)

Explanation:

Step1: Set up the angle equation

From the diagram, we know that \( \angle RST=(6x + 12)^\circ\), and it is composed of \(78^\circ\) and \((3x - 12)^\circ\). So we can set up the equation: \(6x+12=78 + 3x-12\).

Step2: Solve for \(x\)

Subtract \(3x\) from both sides: \(6x - 3x+12=78 + 3x-3x-12\), which simplifies to \(3x + 12=66\). Then subtract 12 from both sides: \(3x+12 - 12=66 - 12\), so \(3x = 54\). Divide both sides by 3: \(x=\frac{54}{3}=18\).

Step3: Calculate \(m\angle RST\)

Substitute \(x = 18\) into the expression \((6x + 12)^\circ\). So \(6\times18+12=108 + 12=120^\circ\).

Answer:

\(120^\circ\) (corresponding to the option "120°")