Question

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

1. click here to insert 5. transitive property
2. click here to insert 6. transitive property
3. click here to insert 7. click here to insert
4. \\( \triangle s p t \\) is isosceles. 8. definition of an isosceles triangle

Explanation:

Step1: Use the property of isosceles triangle

Since \( \overline{PQ}\cong\overline{PR}\), then \( \angle3=\angle4\) (Base - angles of an isosceles triangle are equal).

Step2: Use the property of parallel lines

Because \( \overline{QR}\parallel\overline{ST}\), so \( \angle1 = \angle3\) (Alternate - interior angles) and \( \angle2=\angle4\) (Alternate - interior angles).

Step3: Use the transitive property

By the transitive property, \( \angle1=\angle2\) (Since \( \angle1 = \angle3\), \( \angle2=\angle4\) and \( \angle3=\angle4\)).

Step4: Use the definition of isosceles triangle

In \( \triangle SPT\), if \( \angle1=\angle2\), then \( \overline{PS}\cong\overline{PT}\) (Converse of the base - angles theorem). And by the definition of an isosceles triangle (a triangle with at least two congruent sides), \( \triangle SPT\) is isosceles.

Answer:

1. \( \angle3=\angle4\)
2. \( \angle1 = \angle3\), \( \angle2=\angle4\)
3. \( \angle1=\angle2\); If two angles of a triangle are equal, then the sides opposite those angles are equal (Converse of the base - angles theorem)