Question

consider the following figure.
(note that the figure is not drawn to scale.)
order the side lengths df, fg, de, dg, and ge from least to greatest.
□ < □ < □ < □ < □

Explanation:

Step1: Analyze Triangle \( \triangle DEG \)

In \( \triangle DEG \), \( \angle E = 28^\circ \), \( \angle EDG = 29^\circ \), so \( \angle DGE = 180^\circ - 28^\circ - 29^\circ = 123^\circ \). By the triangle side - angle relationship (larger angle opposite longer side), in \( \triangle DEG \), \( \angle E < \angle EDG < \angle DGE \), so \( DG < DE < EG \)? Wait, no, wait: angle - side relationship is that the larger the angle, the longer the side opposite to it. So in \( \triangle DEG \), side opposite \( \angle E \) is \( DG \), side opposite \( \angle EDG \) is \( EG \), side opposite \( \angle DGE \) is \( DE \). So \( \angle E = 28^\circ \), \( \angle EDG = 29^\circ \), \( \angle DGE = 123^\circ \). So \( DG \) (opposite \( 28^\circ \)) \( < EG \) (opposite \( 29^\circ \)) \( < DE \) (opposite \( 123^\circ \))? Wait, no, \( \angle E = 28^\circ \) (opposite \( DG \)), \( \angle EDG = 29^\circ \) (opposite \( EG \)), \( \angle DGE = 123^\circ \) (opposite \( DE \)). So since \( 28^\circ<29^\circ < 123^\circ \), then \( DG < EG < DE \).

Step2: Analyze Triangle \( \triangle DFG \)

In \( \triangle DFG \), \( \angle F = 84^\circ \), \( \angle FDG \): since \( \angle EDG = 29^\circ \), and assuming \( \angle EDF \) is a straight line? Wait, no, the figure has \( G \) on \( EF \). So \( \angle DGF \): since \( \angle DGE = 123^\circ \), then \( \angle DGF = 180^\circ - 123^\circ = 57^\circ \). Then in \( \triangle DFG \), \( \angle F = 84^\circ \), \( \angle DGF = 57^\circ \), so \( \angle FDG = 180^\circ - 84^\circ - 57^\circ = 39^\circ \). By angle - side relationship, in \( \triangle DFG \), side opposite \( \angle F = 84^\circ \) is \( DG \), side opposite \( \angle DGF = 57^\circ \) is \( DF \), side opposite \( \angle FDG = 39^\circ \) is \( FG \). So \( FG \) (opposite \( 39^\circ \)) \( < DF \) (opposite \( 57^\circ \)) \( < DG \) (opposite \( 84^\circ \))? Wait, no: angle - side: larger angle opposite longer side. So \( \angle FDG = 39^\circ \), \( \angle DGF = 57^\circ \), \( \angle F = 84^\circ \). So side opposite \( 39^\circ \) is \( FG \), side opposite \( 57^\circ \) is \( DF \), side opposite \( 84^\circ \) is \( DG \). So \( FG < DF < DG \).

Step3: Combine the relationships

From \( \triangle DFG \): \( FG < DF < DG \)
From \( \triangle DEG \): \( DG < EG < DE \) (Wait, no, earlier mistake: in \( \triangle DEG \), \( \angle E = 28^\circ \) (opposite \( DG \)), \( \angle EDG = 29^\circ \) (opposite \( EG \)), \( \angle DGE = 123^\circ \) (opposite \( DE \)). So \( DG \) (opposite \( 28^\circ \)) \( < EG \) (opposite \( 29^\circ \)) \( < DE \) (opposite \( 123^\circ \))? But from \( \triangle DFG \), \( FG < DF < DG \). So combining: \( FG < DF < DG < EG < DE \)? Wait, no, let's re - check.

Wait, in \( \triangle DFG \): angles are \( \angle F = 84^\circ \), \( \angle DGF = 57^\circ \), \( \angle FDG = 39^\circ \). So sides: \( FG \) (opposite \( 39^\circ \)) is the shortest, then \( DF \) (opposite \( 57^\circ \)), then \( DG \) (opposite \( 84^\circ \)).

In \( \triangle DEG \): angles are \( \angle E = 28^\circ \), \( \angle EDG = 29^\circ \), \( \angle DGE = 123^\circ \). So sides: \( DG \) (opposite \( 28^\circ \)), \( EG \) (opposite \( 29^\circ \)), \( DE \) (opposite \( 123^\circ \)). So \( DG < EG < DE \).

Now, since \( FG < DF < DG \) and \( DG < EG < DE \), we can chain them: \( FG < DF < DG < EG < DE \)? Wait, but the sides to order are \( DF \), \( FG \), \( DE \), \( DG \), \( GE \) (which is \( EG \)).

So from \( \triangle DFG \): \( FG < DF < DG \)

From \( \triangle DEG \): \( DG < GE < DE \)

So combining all: \( FG < DF < DG < GE < DE \)

Answer:

\( FG < DF < DG < GE < DE \)