Question

complete the proof that ( \triangle qtucong\triangle rsu ).
1 ( overline{tu}congoverline{su} )
2 ( overline{qu}congoverline{ru} )
3 ( angle qutcongangle rus )
4 ( \triangle qtucong\triangle rsu )

Explanation:

Step1: Given

From the figure, we can directly observe that \( \overline{TU}\) and \( \overline{SU}\) are marked as congruent. So, the reason for \( \overline{TU}\cong\overline{SU}\) is "Given".

Step2: Given

Similarly, \( \overline{QU}\) and \( \overline{RU}\) are marked as congruent in the figure. So, the reason for \( \overline{QU}\cong\overline{RU}\) is "Given".

Step3: Vertical Angles Theorem

\( \angle QUT\) and \( \angle RUS\) are vertical angles. By the Vertical Angles Theorem, vertical angles are congruent. So, the reason for \( \angle QUT\cong\angle RUS\) is "Vertical Angles Theorem".

Step4: SAS (Side - Angle - Side) Congruence Criterion

We have two sides (\( \overline{TU}\cong\overline{SU}\), \( \overline{QU}\cong\overline{RU}\)) and the included angle (\( \angle QUT\cong\angle RUS\)) congruent. By the SAS (Side - Angle - Side) congruence criterion for triangles, \( \triangle QTU\cong\triangle RSU\). So, the reason for \( \triangle QTU\cong\triangle RSU\) is "SAS (Side - Angle - Side)".

Answer:

1. Given
2. Given
3. Vertical Angles Theorem
4. SAS (Side - Angle - Side)