Question

1. if ( mangle rst=(12x - 1)^{circ}), ( mangle rsu=(9x - 15)^{circ}), and ( mangle ust = 53^{circ}), find each measure.

( x=)
( mangle rst=)
( mangle rsu=)

Explanation:

Step1: Use the angle - addition postulate

The angle - addition postulate states that \(m\angle RST=m\angle RSU + m\angle UST\).
Substitute the given expressions: \(12x - 1=(9x - 15)+53\).

Step2: Solve the equation for \(x\)

Simplify the right - hand side of the equation: \(12x - 1=9x+38\).
Subtract \(9x\) from both sides: \(12x-9x - 1=9x - 9x+38\), which gives \(3x - 1=38\).
Add \(1\) to both sides: \(3x-1 + 1=38 + 1\), so \(3x=39\).
Divide both sides by \(3\): \(x=\frac{39}{3}=13\).

Step3: Find \(m\angle RST\)

Substitute \(x = 13\) into the expression for \(m\angle RST\): \(m\angle RST=12x-1\).
\(m\angle RST=12\times13 - 1=156 - 1=155^{\circ}\).

Step4: Find \(m\angle RSU\)

Substitute \(x = 13\) into the expression for \(m\angle RSU\): \(m\angle RSU=9x - 15\).
\(m\angle RSU=9\times13-15=117 - 15=102^{\circ}\).

Answer:

\(x = 13\)
\(m\angle RST=155^{\circ}\)
\(m\angle RSU = 102^{\circ}\)