Question

in δghi, \\(\overline{gi}\\) is extended through point i to point j, \\(m\angle hij = (9x - 15)\degree\\), \\(m\angle igh = (3x + 13)\degree\\), and \\(m\angle ghi = (3x + 8)\degree\\). find \\(m\angle igh\\).

Explanation:

Step1: Use 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 $\triangle GHI$, $\angle HIJ$ is an exterior angle, so $m\angle HIJ=m\angle IGH + m\angle GHI$.
Substitute the given angle measures into the equation:
$$9x - 15=(3x + 13)+(3x + 8)$$

Step2: Simplify the right - hand side of the equation

Combine like terms on the right - hand side:
$$9x-15 = 3x+3x + 13 + 8$$
$$9x-15=6x + 21$$

Step3: Solve for x

Subtract $6x$ from both sides:
$$9x-6x-15=6x - 6x+21$$
$$3x-15 = 21$$
Add 15 to both sides:
$$3x-15 + 15=21 + 15$$
$$3x=36$$
Divide both sides by 3:
$$x=\frac{36}{3}=12$$

Step4: Find $m\angle IGH$

We know that $m\angle IGH=(3x + 13)^{\circ}$. Substitute $x = 12$ into the expression:
$$m\angle IGH=3\times12+13$$
$$m\angle IGH = 36+13$$
$$m\angle IGH=49$$

Answer:

$49$