# Thread: Modern Geometry: Proving Angles Congruent

1. ## 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 = mWTS by CPCF (Congruent Parts of Congruent Figures)

m∠WSR + mWST = 180, and m∠WTU + mWTS = 180.

m∠
WSR = mWTU 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.

2. BIG HINT: Use the external angle theorem.
$\displaystyle m\left( {\angle WST} \right) = m\left( {\angle WRS} \right) + m\left( {\angle SWR} \right)\;\& \,m\left( {\angle WTS} \right) = m\left( {\angle WUT} \right) + m\left( {\angle TWU} \right)$

3. ## Ok

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
$\displaystyle {\angle WST}$ = $\displaystyle {\angle WTS}$ since triangle WST is isosceles with WS = WT. So, since $\displaystyle {\angle WST}$ = $\displaystyle {\angle WTS}$, 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, $\displaystyle {\angle WSR}$ = $\displaystyle {\angle WTU}$, and RS = TU), triangles WSR and WTU are congruent; so $\displaystyle {\angle RWS}$ and $\displaystyle {\angle TWU}$ are also congruent by CPCF.