Question

in \\( \triangle efg, m \angle e=(7 x+12)^{\circ}, m \angle f=(6 x-9)^{\circ} \\), and \\( m \angle g=(2 x+12)^{\circ} \\). what is the value of \\( x \\)?

Explanation:

Step1: Use the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(m\angle E+m\angle F + m\angle G=180^{\circ}\).
Substitute the given angle expressions: \((7x + 12)+(6x-9)+(2x + 12)=180\).

Step2: Combine like terms

First, combine the \(x\) - terms: \(7x+6x + 2x=15x\).
Then, combine the constant terms: \(12-9 + 12=15\).
The equation becomes \(15x+15 = 180\).

Step3: Solve for \(x\)

Subtract \(15\) from both sides of the equation: \(15x+15-15=180 - 15\), which simplifies to \(15x=165\).
Divide both sides by \(15\): \(x=\frac{165}{15}\).

Answer:

\(x = 11\)