Thread: Modern Geometry: Proving Angles Congruent

If WS = WT and RS = ST = TU, with R-S-T-U, prove that ∠RWS ∠TWU.

My work so far:
Proof:
Given: WS = WT and RS = ST = TU, with R-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, ∠RWS ∠TWU by CPCF.

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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.

BIG HINT: Use the external angle theorem.

Originally Posted by Plato

BIG HINT: Use the external angle theorem.

My struggles with this class (Modern Geometry) is that it's like we basically have to forget everything we learned in our high school geometry class because we are "building up our repertoire" of tools to use for proofs. So in this case, I don't know if we can use the external angle theorem. (Our text is David Kay's College Geometry 2nd ed.)

But anyway, if I can use the external angle theorem, I already know
= since triangle WST is isosceles with WS = WT. So, since = , then wouldn't their complements be equal by the theorem that "2 angles that are supplementary to the same angle have equal measures." And thus by, SAS (WS = WT, = , and RS = TU), triangles WSR and WTU are congruent; so and are also congruent by CPCF.