Question

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

answer attempt 1 out of 2

$\text{m}\angle gef = \square\degree$

Explanation:

Step1: Apply 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 the two non - adjacent interior angles are $\angle EFG$ and $\angle GEF$. So we have the equation:
$m\angle FGH=m\angle EFG + m\angle GEF$
Substitute the given angle measures:
$(9x - 15)=(3x + 15)+(2x+6)$

Step2: Simplify and Solve for x

First, simplify the right - hand side of the equation:
$(3x + 15)+(2x + 6)=3x+2x+15 + 6=5x+21$
So our equation becomes:
$9x-15 = 5x+21$
Subtract $5x$ from both sides:
$9x-5x-15=5x - 5x+21$
$4x-15 = 21$
Add 15 to both sides:
$4x-15 + 15=21 + 15$
$4x=36$
Divide both sides by 4:
$x=\frac{36}{4}=9$

Step3: Find $m\angle GEF$

We know that $m\angle GEF=(2x + 6)^{\circ}$. Substitute $x = 9$ into the expression:
$m\angle GEF=2\times9+6=18 + 6=24$

Answer:

$24$