Question

1. if ( k ) is the incenter of ( \triangle efg ), find ( x ) and each angle measure.

( x = )
( mangle kej = )
( mangle efg = )
( mangle fge = )
( mangle kgj = )

Explanation:

Step1: Use incenter angle property

The incenter creates angle bisectors, so $\angle FEK = \angle GEK$. Set the expressions equal:
$$18x - 23 = 7x - 1$$

Step2: Solve for x

Subtract $7x$ from both sides, add 23 to both sides:
$$18x - 7x = -1 + 23$$
$$11x = 22$$
$$x = \frac{22}{11} = 2$$

Step3: Calculate $m\angle KEJ$

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

Step4: Calculate $m\angle EFG$

$K$ is the incenter, so $FK$ bisects $\angle EFG$. Given $\angle IFG=47^\circ$, so:
$$m\angle EFG = 2 \times 47^\circ = 94^\circ$$

Step5: Calculate $\angle FEG$ first

$\angle FEG = 2 \times m\angle KEJ = 2 \times 13^\circ = 26^\circ$

Step6: Calculate $m\angle FGE$

Use triangle angle sum ($180^\circ$):
$$m\angle FGE = 180^\circ - 94^\circ - 26^\circ = 60^\circ$$

Step7: Calculate $m\angle KGJ$

$GK$ bisects $\angle FGE$, so:
$$m\angle KGJ = \frac{60^\circ}{2} = 30^\circ$$

Answer:

$x = 2$
$m\angle KEJ = 13^\circ$
$m\angle EFG = 94^\circ$
$m\angle FGE = 60^\circ$
$m\angle KGJ = 30^\circ$