Question

1. given: \\( \overline{st} \parallel \overline{uv}, \overline{st} \cong \overline{vu} \\) prove: \\( \triangle stw \cong \triangle vuw \\)

Explanation:

Step1: Use the property of parallel lines

Since \( \overline{ST}\parallel\overline{UV}\), then \(\angle TSW=\angle VUW\) (alternate - interior angles) and \(\angle STW=\angle VUW\) (alternate - interior angles).

Step2: State the given side - equality

We are given that \( \overline{ST}\cong\overline{VU}\).

Step3: Use the ASA (Angle - Side - Angle) congruence criterion

In \(\triangle STW\) and \(\triangle VUW\), we have \(\angle TSW=\angle VUW\) (from Step 1), \( \overline{ST}\cong\overline{VU}\) (given), and \(\angle STW=\angle VUW\) (from Step 1). By the ASA congruence criterion, \(\triangle STW\cong\triangle VUW\).

Answer:

1. \( \overline{ST}\parallel\overline{UV}\), \( \overline{ST}\cong\overline{VU}\) (Given)
2. \(\angle TSW=\angle VUW\), \(\angle STW=\angle VUW\) (Alternate - interior angles for parallel lines \( \overline{ST}\parallel\overline{UV}\))
3. \(\triangle STW\) and \(\triangle VUW\) (Triangles to be compared)
4. ASA (Angle - Side - Angle) congruence criterion (\(\angle TSW=\angle VUW\), \( \overline{ST}\cong\overline{VU}\), \(\angle STW=\angle VUW\))
5. \(\triangle STW\cong\triangle VUW\) (Conclusion from ASA)