Question

in \\( \triangle uvw, m \angle u=(x + 11)^{\circ}, m \angle v=(4 x-10)^{\circ}, \\) and \\( m \angle w=(9 x+11)^{\circ} \\). find \\( m \angle u \\).

Explanation:

Step1: Recall triangle angle sum theorem

The sum of the interior angles of a triangle is \(180^\circ\). So, \(m\angle U + m\angle V + m\angle W = 180^\circ\).
Substitute the given angle measures: \((x + 11) + (4x - 10) + (9x + 11) = 180\).

Step2: Simplify the left - hand side

Combine like terms: \(x+4x + 9x+11-10 + 11=180\).
\(14x+12 = 180\).

Step3: Solve for x

Subtract 12 from both sides: \(14x=180 - 12=168\).
Divide both sides by 14: \(x=\frac{168}{14} = 12\).

Step4: Find \(m\angle U\)

Substitute \(x = 12\) into the expression for \(m\angle U\): \(m\angle U=(12 + 11)^\circ=23^\circ\).

Answer:

\(23^\circ\)