Question

if k is the incenter of the triangle, find x and the measure of the following angles.
(18x - 23)°
(7x - 1)°
47°
x= type your answer...
<gef= type your answer...
<fgk= type your answer...
<efk= type your answer...

Explanation:

Step1: Set angles equal (incenter bisects angles)

$18x - 23 = 7x - 1$

Step2: Solve for x

$18x - 7x = 23 - 1$
$11x = 22$
$x = \frac{22}{11} = 2$

Step3: Calculate $\angle GEF$

Substitute $x=2$ into $7x-1$:
$7(2) - 1 = 14 - 1 = 13^\circ$

Step4: Calculate $\angle FGK$

Incenter bisects $\angle F$, so $\angle FGK = \angle IFK = 47^\circ$

Step5: Calculate $\angle EFK$

Incenter bisects $\angle F$, so $\angle EFK = \angle IFK = 47^\circ$

Answer:

$x=2$
$\angle GEF=13^\circ$
$\angle FGK=47^\circ$
$\angle EFK=47^\circ$