Question

le congruence
the proof that \\( \overline { u x } \cong \overline { s v } \\) is shown.
given: \\( \triangle s t u \\) an equilateral triangle
\\( \angle t x u \cong \angle t v s \\)
prove: \\( \overline { u x } \cong \overline { s v } \\)
\\( \

$$\begin{array} { | l | l | } \\hline 4. \\overline { s t } \\cong \\overline { u t } & 4. \\text { sides of an equilat. } \\triangle \\text { are } \\cong \\\\ \\hline 5. ? & 5. \\text { aac } \\\\ \\hline 6. \\overline { u x } \\cong \\overline { s v } & 6. \\text { cpctc } \\\\ \\hline \\end{array}$$

\\)

Explanation:

Step1: Recall the AAS (Angle - Angle - Side) congruence criterion

The AAS congruence criterion states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent.

Step2: Identify the elements for congruence

We know that \(\triangle STU\) is an equilateral triangle, so \(ST = UT\) (given in step 4). We are also given that \(\angle TXU\cong\angle TVS\). And \(\angle T\) is a common angle for \(\triangle TUX\) and \(\triangle TSV\). So, by AAS (\(\angle T\cong\angle T\), \(\angle TXU\cong\angle TVS\), \(UT = ST\)), \(\triangle TUX\cong\triangle TSV\)

Answer:

\(\triangle TUX\cong\triangle TSV\)