Question

given: p || r
prove: ∠2 and ∠5 are supplementary

1. p || r - given
2. ∠1 ≅ ∠5 - corresponding angles thm
3. ∠1 and ∠2 are a linear pair. - def of linear pair
4. ∠1 and ∠2 are supplementary - substitution+
5. m∠1 + m∠2 = 180° - def of supp. angles (1st use)
6. m∠5 + m∠2 = 180° - def of supp. angles (2nd use)
7. ∠2 and ∠5 are supplementary - supplement thm

Explanation:

Step1: State given information

Given \(p\parallel r\)

Step2: Use corresponding - angles theorem

Since \(p\parallel r\), by the Corresponding Angles Theorem, \(\angle1\cong\angle5\), so \(m\angle1 = m\angle5\)

Step3: Define linear - pair

\(\angle1\) and \(\angle2\) are a linear pair by the definition of a linear pair.

Step4: Determine supplementary angles

By the definition of supplementary angles (since \(\angle1\) and \(\angle2\) are a linear pair), \(\angle1\) and \(\angle2\) are supplementary, so \(m\angle1 + m\angle2=180^{\circ}\)

Step5: Substitute \(\angle1\) with \(\angle5\)

Since \(m\angle1 = m\angle5\), substituting \(m\angle1\) in \(m\angle1 + m\angle2 = 180^{\circ}\), we get \(m\angle5 + m\angle2=180^{\circ}\)

Step6: Conclude supplementary angles

By the definition of supplementary angles, since \(m\angle5 + m\angle2 = 180^{\circ}\), \(\angle2\) and \(\angle5\) are supplementary.

Answer:

The proof that \(\angle2\) and \(\angle5\) are supplementary is completed as shown above.