Question

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question
in δefg, eg is extended through point g to point h, m∠efg = (3x + 15)°, m∠gef = (2x + 6)°, and m∠fgh = (9x − 15)°. find m∠gef.

Explanation:

Step1: Recall 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 $\triangle EFG$, $\angle FGH$ is an exterior angle, so $m\angle FGH=m\angle EFG + m\angle GEF$.

Substitute the given angle measures into the formula:
$(9x - 15)=(3x + 15)+(2x+6)$

Step2: Simplify the right - hand side of the equation

Simplify $(3x + 15)+(2x + 6)$:
$3x+2x+15 + 6=5x+21$

So our equation becomes $9x-15 = 5x+21$

Step3: Solve for $x$

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:
$x=\frac{36}{4}=9$

Step4: Find $m\angle GEF$

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

Answer:

$24$