Question

1. complete the proof.

given: \\( \overline { p q } \parallel \overline { s t } ; \overline { p r } \cong \overline { r t } \\)
prove: \\( \triangle p q r \cong \triangle t s r \\)
statements

1. \\( \overline { p q } \parallel \overline { s t } ; \overline { p r } \cong \overline { r t } \\) 1. given
2. \\( \angle r t s \cong \angle r p q \\) 2.
3. \\( \angle t r s \cong \angle p r q \\) 3.
4. \\( \triangle p q r \cong \triangle t s r \\) 4.

(hint: \\( \overline { t p } \\) is a transversal)

Explanation:

Step1: Alternate interior angles

When two parallel lines ($\overline{PQ}\parallel\overline{ST}$) are cut by a transversal ($\overline{TP}$), alternate - interior angles are congruent. So, $\angle RTS\cong\angle RPQ$ (by the Alternate Interior Angles Theorem).

Step2: Vertical angles

Vertical angles are congruent. $\angle TRS$ and $\angle PRQ$ are vertical angles. So, $\angle TRS\cong\angle PRQ$ (by the Vertical Angles Theorem).

Step3: ASA criterion

We have $\overline{PR}\cong\overline{RT}$ (given), $\angle RTS\cong\angle RPQ$ (from step 1), and $\angle TRS\cong\angle PRQ$ (from step 2). So, $\triangle PQR\cong\triangle TSR$ by the Angle - Side - Angle (ASA) congruence criterion.

Answer:

1. Alternate Interior Angles Theorem; 3. Vertical Angles Theorem; 4. ASA (Angle - Side - Angle)