Question

analyzing triangle relationships
analyze the diagram.
which statements regarding the diagram are correct? choose four correct
answers.
$\triangle rstcong\triangle ust$ by sas.
$\triangle rwscong\triangle uwt$ by aas.
$angle wtucongangle wsr$ because cpctc.
$angle rwscongangle uwt$ because they are vertical
angles.

Explanation:

Step1: Check $\triangle RST\cong\triangle UST$ by SAS

In $\triangle RST$ and $\triangle UST$, $RS = US$ (given by the mark), $\angle RST=\angle UST$ (common angle), $ST = ST$ (common side). So $\triangle RST\cong\triangle UST$ by SAS.

Step2: Check $\triangle RWS\cong\triangle UWT$ by AAS

In $\triangle RWS$ and $\triangle UWT$, $\angle RSW=\angle USW$ (from $\triangle RST\cong\triangle UST$), $\angle RWS=\angle UWT$ (vertical angles), $RS = US$. So $\triangle RWS\cong\triangle UWT$ by AAS.

Step3: Check $\angle WTU\cong\angle WSR$ because CPCTC

Since $\triangle RST\cong\triangle UST$, by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), $\angle WTU\cong\angle WSR$.

Step4: Check $\angle RWS\cong\angle UWT$ because they are vertical angles

Vertical angles are always congruent. So $\angle RWS\cong\angle UWT$.

Answer:

$\triangle RST\cong\triangle UST$ by SAS, $\triangle RWS\cong\triangle UWT$ by AAS, $\angle WTU\cong\angle WSR$ because CPCTC, $\angle RWS\cong\angle UWT$ because they are vertical angles.