Question

finding unknown measures
point x is the incenter of \\( \triangle a b c \\).
if \\( e x = 4 z + 1, x f = 2 z + 7 \\), and \\( m \angle a b c = 44 ^ { \circ } \\), find
the following measures.
\\( g x = \\)
\\( m \angle a b x = \\)\\( ^ { \circ } \\)

Explanation:

Step1: Find the value of z

Since X is the incenter, the distances from X to the sides of the triangle are equal. So, \( EX = XF \).
Set \( 4z + 1 = 2z + 7 \).
Subtract \( 2z \) from both sides: \( 4z - 2z + 1 = 2z - 2z + 7 \), which simplifies to \( 2z + 1 = 7 \).
Subtract 1 from both sides: \( 2z + 1 - 1 = 7 - 1 \), so \( 2z = 6 \).
Divide both sides by 2: \( z = \frac{6}{2} = 3 \).

Step2: Find EX (and thus GX)

Substitute \( z = 3 \) into \( EX = 4z + 1 \): \( EX = 4(3) + 1 = 12 + 1 = 13 \).
Since \( GX \) is also a distance from the incenter to a side, \( GX = EX = 13 \).

Step3: Find \( m\angle ABX \)

The incenter is the intersection of the angle bisectors. So, \( BX \) bisects \( \angle ABC \).
Thus, \( m\angle ABX = \frac{1}{2}m\angle ABC \).
Given \( m\angle ABC = 44^\circ \), then \( m\angle ABX = \frac{1}{2}(44^\circ) = 22^\circ \).

Answer:

\( GX = 13 \)
\( m\angle ABX = 22^\circ \)