Question

in $\triangle ghi$, $overline{gi}$ is extended through point i to point j, $mangle igh=(x + 18)^{circ}$, $mangle ghi=(3x + 8)^{circ}$, and $mangle hij=(6x + 6)^{circ}$. find $mangle igh$.

Explanation:

Step1: Use the exterior - angle property

The exterior - angle of a triangle is equal to the sum of the two non - adjacent interior angles. In \(\triangle GHI\), \(\angle HIJ\) is an exterior angle, so \(m\angle HIJ=m\angle IGH + m\angle GHI\).
Substitute the given angle expressions: \((6x + 6)=(x + 18)+(3x + 8)\).

Step2: Simplify the right - hand side of the equation

Combine like terms on the right - hand side: \((x + 18)+(3x + 8)=x+3x + 18 + 8=4x+26\).
So the equation becomes \(6x + 6=4x+26\).

Step3: Solve for \(x\)

Subtract \(4x\) from both sides: \(6x-4x + 6=4x-4x+26\), which simplifies to \(2x+6 = 26\).
Then subtract 6 from both sides: \(2x+6 - 6=26 - 6\), getting \(2x=20\).
Divide both sides by 2: \(x = 10\).

Step4: Find \(m\angle IGH\)

Substitute \(x = 10\) into the expression for \(m\angle IGH\). Since \(m\angle IGH=(x + 18)^{\circ}\), then \(m\angle IGH=(10 + 18)^{\circ}=28^{\circ}\).

Answer:

\(28^{\circ}\)