Question

1. given: \\( \angle 1 \cong \angle 5 ; \angle 3 \cong \angle 4 \\) prove: \\( t \parallel v \\)

Explanation:

Step1: Use corresponding angles converse

Since \(\angle1\cong\angle5\), by the corresponding angles converse, \(r\parallel s\).

Step2: Use vertical angles

\(\angle2\cong\angle3\) (vertical angles are congruent).

Step3: Use substitution

Because \(\angle3\cong\angle4\) (given) and \(\angle2\cong\angle3\), then \(\angle2\cong\angle4\) (transitive property).

Step4: Use alternate - interior angles converse

Since \(\angle2\cong\angle4\), by the alternate - interior angles converse, \(t\parallel v\).

Answer:

Statements	Reasons
2. \(\angle3\cong\angle4\)	2. Given
3. \(r\parallel s\)	3. Corresponding angles converse
4. \(\angle2\cong\angle3\)	4. Vertical angles
5. \(\angle2\cong\angle4\)	5. Transitive property
6. \(t\parallel v\)	6. Alternate - interior angles converse