Question

consider the following figure. (note that the figure is not drawn to scale.) order the side - lengths fg, gi, fh, fi, and ih from least to greatest.

Explanation:

Step1: Find angles in triangles

In $\triangle FGI$, $\angle GFI = 44^{\circ}$, $\angle FGI=89^{\circ}$, so $\angle FIG=180^{\circ}-44^{\circ}-89^{\circ}=47^{\circ}$. In $\triangle FIH$, $\angle IFH = 180^{\circ}-44^{\circ}=136^{\circ}$, $\angle FIH = 180^{\circ}-\angle FIG=133^{\circ}$, $\angle FHI = 11^{\circ}$, so $\angle FIH = 133^{\circ}$, $\angle IFH = 136^{\circ}$, $\angle FHI = 11^{\circ}$.

Step2: Apply angle - side relationship

In a triangle, the side opposite the smallest angle is the shortest and the side opposite the largest angle is the longest.
In $\triangle FGI$, since $44^{\circ}<47^{\circ}<89^{\circ}$, $GI < FG< FI$.
In $\triangle FIH$, since $11^{\circ}<133^{\circ}<136^{\circ}$, $FHCombining the two triangles, comparing all the sides $IH

Answer:

$IH