Question

in \\( \triangle efg \\), \\( overline{eg} \\) is extended through point \\( g \\) to point \\( h \\), \\( m\angle efg=(3x + 15)^{circ} \\), \\( m\angle gef=(2x + 6)^{circ} \\), and \\( m\angle fgh=(9x - 15)^{circ} \\). find \\( m\angle gef \\).

Explanation:

Step1: Apply the exterior - angle theorem

The exterior - angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles.
In \(\triangle EFG\), \(\angle FGH\) is an exterior angle, and \(\angle EFG\) and \(\angle GEF\) are the non - adjacent interior angles. So, \(m\angle FGH=m\angle EFG + m\angle GEF\).
Substitute the given angle measures: \((9x - 15)=(3x + 15)+(2x + 6)\).

Step2: Solve the equation for \(x\)

Simplify the right - hand side of the equation: \((3x + 15)+(2x + 6)=3x+2x + 15 + 6=5x+21\).
The equation becomes \(9x - 15=5x+21\).
Subtract \(5x\) from both sides: \(9x-5x - 15=5x-5x + 21\), which gives \(4x-15 = 21\).
Add \(15\) to both sides: \(4x-15 + 15=21 + 15\), so \(4x=36\).
Divide both sides by \(4\): \(x=\frac{36}{4}=9\).

Step3: Find \(m\angle GEF\)

Substitute \(x = 9\) into the formula for \(m\angle GEF\).
Since \(m\angle GEF=(2x + 6)^{\circ}\), then \(m\angle GEF=(2\times9 + 6)^{\circ}\).
First, calculate \(2\times9=18\), then \(18 + 6=24\).

Answer:

\(24^{\circ}\)