Answer:
IH < GI < GH < FH < FG
Explanation:
Given quadrilateral FGIH and diagonal GH with interior angles of the triangles marked, you want to know the order of the lengths of the line segments from least to greatest.
Missing angles
The two missing angles can be found from the angle sum theorem, which tells you the sum of angles in a triangle is 180°.
∠FHG = 180° -∠F -∠G
∠FHG = 180° -51° -56° = 73°
and
∠I = 180° -∠IGH -∠IHG
∠I = 180° -54° -62° = 64°
Angle ordering
In a triangle, the order of side lengths is the same as the order of opposite angle measures. For the two given triangles, this tells us ...
∠F < ∠FGH < ∠FHG ⇒ GH < FH < FG
and
∠IGH < ∠IHG < ∠I ⇒ IH < GI < GH
We note that segment GH is at the extreme of each of these partial orders, so we can combine them to give the segments in least-to-greatest order.
From least to greatest, the side lengths are IH < GI < GH < FH < FG.
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Additional comment
The attachments show the relative numerical values of the side lengths with side GH taken a 1 unit.
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