Question

given: p||t and q||r
prove: ∠1≅∠3
statements reasons
p||t and q||r given

Explanation:

Step1: Use parallel line property

Since \(p\parallel t\), by the alternate - interior angles theorem, \(\angle1\cong\angle2\).

Step2: Use parallel line property again

Since \(q\parallel r\), by the vertical angles theorem (or alternate - interior angles considering the transversal), \(\angle2\cong\angle3\).

Step3: Use transitive property

If \(\angle1\cong\angle2\) and \(\angle2\cong\angle3\), then by the transitive property of congruence (\(a = b\) and \(b = c\) implies \(a = c\)), \(\angle1\cong\angle3\).

Answer:

Statements	Reasons
\(\angle1\cong\angle2\)	Alternate - interior angles theorem (\(p\parallel t\))
\(\angle2\cong\angle3\)	Vertical angles theorem (or alternate - interior angles considering \(q\parallel r\))
\(\angle1\cong\angle3\)	Transitive property of congruence