Question

in \\( \triangle tuv, m \angle t=(7 x+10)^{\circ}, m \angle u=(x+2)^{\circ} \\), and \\( m \angle v=(3 x+3)^{\circ} \\). find \\( m \angle u \\).

Explanation:

Step1: Use the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(m\angle T+m\angle U + m\angle V=180^{\circ}\).
Substitute the given angle expressions: \((7x + 10)+(x + 2)+(3x + 3)=180\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \((7x+x + 3x)+(10 + 2+3)=180\).
\(11x+15 = 180\).

Step3: Solve for \(x\)

Subtract \(15\) from both sides: \(11x=180 - 15\).
\(11x=165\).
Divide both sides by \(11\): \(x=\frac{165}{11}=15\).

Step4: Find \(m\angle U\)

Substitute \(x = 15\) into the expression for \(m\angle U\).
\(m\angle U=(x + 2)^{\circ}=(15+2)^{\circ}=17^{\circ}\).

Answer:

\(14^{\circ}\)