Question

in \\( \triangle hij, m \angle h=(4 x-9)^{circ}, m \angle i=(3 x-17)^{circ} \\), and \\( m \angle j=(3 x+16)^{circ} \\). find \\( m \angle j \\).

Explanation:

Step1: Use the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(m\angle H+m\angle I + m\angle J=180^{\circ}\).
Substitute the given angle expressions: \((4x - 9)+(3x - 17)+(3x + 16)=180\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \((4x+3x + 3x)+(-9-17 + 16)=180\).
\(10x-10 = 180\).

Step3: Solve for \(x\)

Add \(10\) to both sides of the equation: \(10x-10 + 10=180 + 10\), which gives \(10x=190\).
Divide both sides by \(10\): \(x=\frac{190}{10}=19\).

Step4: Find \(m\angle J\)

Substitute \(x = 19\) into the expression for \(m\angle J\).
\(m\angle J=(3x + 16)^{\circ}\), so \(m\angle J=3\times19+16\).
\(m\angle J=57 + 16=61^{\circ}\).

Answer:

\(61^{\circ}\)