Let's start by highlighting the congruent parts and the given information in the given diagram.

Since $K$ is the midpoint of $\overline{Q S},$ we have that $\overline{Q K} ≅ \overline{K S} .$ Remember that we are told that $m ∠ S K L > m ∠ Q K J .$ Next, we apply the Hinge Theorem to $△ K S L$ and $△ K J Q .$

By adding

R L

to both sides of the latter inequality we get

R L + L S > Q J + R L .

However, since

R L = R J

we get

R L + L S > R J + Q J .

By applying the Segment Addition Postulate we obtain the required inequality.

R L + L S > R J + Q J \Rightarrow R S > R Q ✓

Two-Column Proof

In the following table we summarize the proof we did before.

Statements	Reasons
$\overline{L K} ≅ \overline{J K}$ and $m ∠ S K L > m ∠ Q K J$ and $K$ is the midpoint of $\overline{Q S}$	Given
$\overline{Q K} ≅ \overline{K S}$	Definition of midpoint
$L S > Q J$	Hinge Theorem
$L S + R L > Q J + R L$	Adding $R L$ to both sides
$\overline{R L} ≅ \overline{R J}$	Given
$R L = R J$	Definition of congruent segments
$L S + R L > Q J + R J$	Substitution
$R S > Q R$	Segment Addition Postulate

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