Question

complete the proof that \\( \triangle t u x \cong \triangle v u w \\).

Explanation:

Step1: Identify the given congruent segments

From the figure, we can see that \( \overline{UW}\) and \( \overline{UX}\) are marked as congruent. So the reason for \( \overline{UW}\cong\overline{UX}\) is "Given (marked in the diagram)".

Step2: Identify the given congruent angles

From the figure, \( \angle V\) and \( \angle T\) are marked as congruent. So the reason for \( \angle V\cong\angle T\) is "Given (marked in the diagram)".

Step3: Vertical angles congruence

\( \angle TUX\) and \( \angle VUW\) are vertical angles. By the Vertical Angles Theorem, \( \angle TUX\cong\angle VUW\). The reason is "Vertical Angles Theorem".

Step4: AAS (Angle - Angle - Side) congruence criterion

We have two angles (\( \angle V\cong\angle T\), \( \angle TUX\cong\angle VUW\)) and a non - included side (\( \overline{UW}\cong\overline{UX}\)). By the AAS (Angle - Angle - Side) congruence criterion, \( \triangle TUX\cong\triangle VUW\). The reason is "AAS (Angle - Angle - Side) Congruence Theorem".

Answer:

Statement	Reason
\( \angle V\cong\angle T\)	Given
\( \angle TUX\cong\angle VUW\)	Vertical Angles Theorem
\( \triangle TUX\cong\triangle VUW\)	AAS (Angle - Angle - Side) Congruence Theorem