Question

1. complete the proof with proof blocks or the 2 - column. given: m∠2 + m∠3 = 180°, ∠2 and ∠5 are supplementary. prove: p || q. ∠2 and ∠5 are supplementary, m∠2 + m∠3 = 180°, ∠2 and ∠3 are supplementary, ∠3 ≅ ∠5, p || q

Explanation:

Step1: Recall the definition of supplementary angles

If \(m\angle2 + m\angle3=180^{\circ}\), by the definition of supplementary - angles, \(\angle2\) and \(\angle3\) are supplementary.

Step2: Use the given supplementary - angle relationship

Given that \(\angle2\) and \(\angle5\) are supplementary, so \(m\angle2 + m\angle5 = 180^{\circ}\). Also, \(m\angle2 + m\angle3 = 180^{\circ}\). Then, by the transitive property of equality, \(m\angle2 + m\angle5=m\angle2 + m\angle3\). Subtracting \(m\angle2\) from both sides, we get \(m\angle3=m\angle5\), so \(\angle3\cong\angle5\).

Step3: Apply the corresponding - angles postulate

Since \(\angle3\) and \(\angle5\) are corresponding angles and \(\angle3\cong\angle5\), by the corresponding - angles postulate, if two lines are cut by a transversal and the corresponding angles are congruent, then the two lines are parallel. So \(p\parallel q\).

Answer:

The missing reasons in the proof (from top - to - bottom) are: "Definition of supplementary angles", "Transitive property of equality (or Subtraction property of equality to get \(\angle3\cong\angle5\))", "Corresponding angles postulate".