IfWS=WTandRS=ST=TU, withR-S-T-U, prove that ∠RWS∠TWU.

My work so far:

Proof:

Given:WS=WTandRS=ST=TU, withR-S-T-U

Given that R-S-T-U, we know R-S-T, S-T-U, R-S-U, and R-T-U.

∠WST ∠WTS by the Isosceles Triangle Theorem.

m∠WST = m∠WTS by CPCF (Congruent Parts of Congruent Figures)

m∠WSR + m∠WST = 180, and m∠WTU + m∠WTS = 180.

m∠WSR = m∠WTU by algebra

∠WSR ∠WTU, and therefore,

Triangle WSR Triangle WTU by SAS.

And thus,∠by CPCF.RWS∠TWU

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Am I even close to having a legitimate proof? I'm pretty sure there is something missing since I did not use the given betweenness relation of R-S-T-U.

Any corrections/suggestions, tips/help is greatly appreciated. Thank you very much for your time.