Question

solve for x.
9)
10)
11)

Explanation:

Step1: Use the exterior angle theorem

The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. In triangle \(PQ R\), \(\angle PRH\) is an exterior angle. So, \(\angle PRH=\angle P+\angle Q\).
Given \(\angle P = 30^{\circ}\), \(\angle Q=(4x + 2)^{\circ}\), and \(\angle PRH=(8 + 6x)^{\circ}\).
We can set up the equation: \(8 + 6x=30+(4x + 2)\).

Step2: Simplify the equation

First, simplify the right - hand side of the equation: \(30+(4x + 2)=4x+32\).
So the equation becomes \(8 + 6x=4x+32\).
Subtract \(4x\) from both sides: \(8 + 6x-4x=4x+32-4x\).
We get \(8 + 2x=32\).

Step3: Solve for \(x\)

Subtract \(8\) from both sides: \(2x=32 - 8\).
Since \(32−8 = 24\), the equation is \(2x=24\).
Divide both sides by \(2\): \(x=\frac{24}{2}\).

Answer:

\(x = 12\)