Question

given: \\( \overline { p q } \cong \overline { p r } \\), \\( \overline { q r } \parallel \overline { s t } \\)
prove: \\( \triangle s p t \\) is isosceles.

1. \\( \overline { p q } \cong \overline { p r } \\) 1. given
2. 2. base angles theorem
3. \\( \overline { q r } \parallel \overline { s t } \\) 3. given
4. 4.
5. 5. transitive property

Explanation:

Step1: Apply the Base Angles Theorem

Since \( \overline{PQ}\cong\overline{PR}\), by the Base - Angles Theorem, \( \angle3\cong\angle4\).

Step2: Use the properties of parallel lines

Because \( \overline{QR}\parallel\overline{ST}\), by the Corresponding Angles Postulate, \( \angle1\cong\angle3\) and \( \angle2\cong\angle4\).

Answer:

1. \( \angle3\cong\angle4\)
2. \( \angle1\cong\angle3,\angle2\cong\angle4\); 4. Corresponding Angles Postulate