Question

in \\( \triangle fgh \\), if \\( m \angle g \\) is five less than twice \\( m \angle f \\) and \\( m \angle h \\) is eighteen less than four times \\( m \angle f \\), find the measure of \\( \angle g \\).

Explanation:

Step1: Use the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(m\angle F + m\angle G + m\angle H=180^{\circ}\). Given \(m\angle F = x\), \(m\angle G = 2x - 5\), and \(m\angle H=4x - 18\).
Substitute these into the angle - sum formula: \(x+(2x - 5)+(4x - 18)=180\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \((x + 2x+4x)+(-5 - 18)=180\), which gives \(7x-23 = 180\).

Step3: Solve for \(x\)

Add \(23\) to both sides of the equation: \(7x-23 + 23=180 + 23\), so \(7x=203\).
Divide both sides by \(7\): \(x=\frac{203}{7}=29\).

Step4: Find the measure of \(\angle G\)

Since \(m\angle G = 2x - 5\), substitute \(x = 29\) into the formula. Then \(m\angle G=2\times29-5\).
Calculate \(2\times29-5=58 - 5=53\).

Answer:

\(53^{\circ}\)