Question

in δuvw, m∠u = (x − 6)°, m∠v = (5x + 6)°, and m∠w = (3x + 18)°. what is the value of x?

Explanation:

Step1: Recall triangle angle sum

The sum of angles in a triangle is \(180^\circ\). So, \(m\angle U + m\angle V + m\angle W = 180^\circ\).

Step2: Substitute angle expressions

Substitute \(m\angle U=(x - 6)^\circ\), \(m\angle V=(5x + 6)^\circ\), \(m\angle W=(3x + 18)^\circ\) into the equation:
\((x - 6)+(5x + 6)+(3x + 18)=180\)

Step3: Simplify the left side

Combine like terms: \(x - 6 + 5x + 6 + 3x + 18 = (x + 5x + 3x)+(-6 + 6 + 18)=9x + 18\)

Step4: Solve for \(x\)

Set \(9x + 18 = 180\). Subtract 18 from both sides: \(9x = 180 - 18 = 162\). Divide by 9: \(x=\frac{162}{9}=18\)

Answer:

\(x = 18\)