Question

if k is the incenter of the triangle, find x and the measure of the following angles.
(18x - 3)°
(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)

Since \( K \) is the incenter, it bisects \( \angle FEG \), so \( 18x - 23 = 7x - 1 \).

Step2: Solve for \( x \)

Subtract \( 7x \) from both sides: \( 11x - 23 = -1 \)
Add 23 to both sides: \( 11x = 22 \)
Divide by 11: \( x = \frac{22}{11} = 2 \)

Step3: Calculate \( \angle GEF \)

First find one half: \( 7x - 1 = 7(2) - 1 = 13^\circ \)
Double it for full angle: \( \angle GEF = 2 \times 13^\circ = 26^\circ \)

Step4: Calculate \( \angle FGK \)

Incenter bisects \( \angle EFG \), so \( \angle FGK = \angle IFK = 47^\circ \)

Step5: Calculate \( \angle EFK \)

First find \( \angle HEK = 18x - 23 = 18(2) - 23 = 13^\circ \), and \( \angle HFK = 47^\circ \). In \( \triangle EFK \), use angle sum:
\( \angle EFK = 180^\circ - 90^\circ - 13^\circ = 77^\circ \) (Note: The incenter forms right angles with sides, so \( \angle EHF = 90^\circ \))

Answer:

\( x = 2 \)
\( \angle GEF = 26^\circ \)
\( \angle FGK = 47^\circ \)
\( \angle EFK = 77^\circ \)