Question

m<egf =

m<fgt =

Explanation:

Step1: Find \( m\angle EGF \)

The sum of angles in a triangle is \( 180^\circ \). In \( \triangle EGF \), we know two angles: \( \angle E = 28^\circ \) and \( \angle F = 58^\circ \). So, \( m\angle EGF = 180^\circ - 28^\circ - 58^\circ \).
\( 180 - 28 - 58 = 94^\circ \)

Step2: Find \( m\angle FGT \)

\( \angle EGF \) and \( \angle FGT \) are supplementary (they form a linear pair), so their sum is \( 180^\circ \). Thus, \( m\angle FGT = 180^\circ - m\angle EGF \).
Substituting \( m\angle EGF = 94^\circ \), we get \( 180 - 94 = 86^\circ \)? Wait, no, wait. Wait, actually, \( \angle FGT \) is an exterior angle? Wait, no, let's re - check. Wait, the triangle angle sum: \( \angle E + \angle F + \angle EGF = 180 \), so \( \angle EGF = 180 - 28 - 58 = 94^\circ \). Then, \( \angle FGT \) and \( \angle EGF \) are supplementary? Wait, no, \( \angle EGT \) is a straight line, so \( \angle EGF + \angle FGT = 180^\circ \)? Wait, no, actually, the exterior angle theorem: the exterior angle at \( G \) ( \( \angle FGT \)) is equal to the sum of the two non - adjacent interior angles. So \( \angle FGT=\angle E+\angle F = 28^\circ+58^\circ = 86^\circ \). Wait, my first calculation for \( \angle FGT \) was wrong. Let's correct:

For \( m\angle EGF \):
Sum of angles in triangle: \( \angle E+\angle F+\angle EGF = 180^\circ \)
\( 28 + 58+\angle EGF=180 \)
\( \angle EGF = 180-(28 + 58)=180 - 86 = 94^\circ \)

For \( m\angle FGT \):
Since \( \angle EGF \) and \( \angle FGT \) are supplementary (they are adjacent and form a straight line \( ET \)), \( \angle EGF+\angle FGT = 180^\circ \)
So \( \angle FGT=180 - \angle EGF = 180 - 94 = 86^\circ \). Alternatively, using exterior angle theorem: the exterior angle of a triangle is equal to the sum of the two remote interior angles. So \( \angle FGT=\angle E+\angle F=28 + 58 = 86^\circ \)

Answer:

\( m\angle EGF = 94^\circ \)
\( m\angle FGT = 86^\circ \)