We will show that S and T are the same points.
Since RT is parallel to side XY, this will show that RS is parallel to side XY.
Let's start with writing the proportion that the Side-Splitter Theorem guarantees.
XR/RQ=Y T/TQ
The ratio on the left-hand side also appears in the proportion that is given.
XR/RQ=Y S/SQ
By the Transitive Property of Equality, the ratios on the right-hand sides are also equal.
Y T/TQ=Y S/SQ
According to the Third Property of Proportions, we can add the denominators to the numerators.
Y T+ TQ/TQ=Y S+ SQ/SQ
Since T and S are both on side YQ, we can use the Segment Addition Property in the numerators.
YQ/TQ=YQ/SQ
Notice that the numerators are the same, so the denominators must also be equal.
TQ= SQ
Since T and S are both on side YQ, this means that T and S are the same points.
T= S
Since by construction segment R T is parallel to side XY, this means that R S is parallel to the same side.
We can summarize the steps above in a flow proof.
