Question

1. fill in the missing information for this proof.

given: ∠1 and ∠3 are supplementary.
prove: a ∥ b

statements	reasons
2)	2) definition of linear pair
3)	4) supplements of the same angle are ≅.
4) a ∥ b	5)

Explanation:

Step1: Analyze Statement 2

A linear pair of angles is formed when two adjacent angles are supplementary (their sum is \(180^\circ\)). From the diagram, \(\angle1\) and \(\angle2\) form a linear pair. So, by the definition of a linear pair, \(\angle1\) and \(\angle2\) are supplementary.
Statement 2: \(\angle1\) and \(\angle2\) are supplementary

Step2: Analyze Statement 3

We know \(\angle1\) and \(\angle3\) are supplementary (given) and \(\angle1\) and \(\angle2\) are supplementary (from step 2). By the theorem "Supplements of the same angle are congruent", \(\angle2\cong\angle3\).
Statement 3: \(\angle2\cong\angle3\)

Step3: Analyze Reason 5

To prove \(a\parallel b\), we can use the converse of the alternate interior angles theorem. Since \(\angle2\) and \(\angle3\) are alternate interior angles and they are congruent, lines \(a\) and \(b\) are parallel.
Reason 5: If alternate interior angles are congruent, then the lines are parallel (Converse of Alternate Interior Angles Theorem)

Answer:

1. \(\boldsymbol{\angle1}\) and \(\boldsymbol{\angle2}\) are supplementary
2. \(\boldsymbol{\angle2\cong\angle3}\)
3. Converse of Alternate Interior Angles Theorem