Question

statements reasons

1. line i and line m intersect 1. given
2. ∠1 is supplementary to ∠2 2. linear pair theorem
3. ? 3. ?
4. ∠1 ≅ ∠3 4. congruent supplements theorem

which statement and reason best completes the proof?
a. 3 m∠1 + m∠3 = 180° 3 definition of supplementary angles
b. 3. ∠1 is supplementary to ∠4 3. linear pair theorem
c. 3 m∠1 + m∠2 = 180° 3. definition of supplementary angles
d. 3. ∠3 is supplementary to ∠2 3 linear pair theorem

Explanation:

Step1: Analyze each option

• Option A: \(m\angle1 + m\angle3=180^{\circ}\) is not based on the linear - pair theorem. Linear - pair theorem is about adjacent angles formed by two intersecting lines.
• Option B: \(\angle1\) and \(\angle4\) are not a linear pair in the context of the given intersection of lines \(l\) and \(m\) (assuming the standard intersection - angle relationships).
• Option C: \(m\angle1 + m\angle2 = 180^{\circ}\) is the result of the linear - pair theorem (statement 2), not a new statement for step 3.
• Option D: Since \(\angle3\) and \(\angle2\) form a linear pair (by the intersection of lines \(l\) and \(m\)), by the linear - pair theorem, \(\angle3\) is supplementary to \(\angle2\). Then, since \(\angle1\) is supplementary to \(\angle2\) (statement 2) and \(\angle3\) is supplementary to \(\angle2\) (step 3), by the congruent supplements theorem (\(\angle1\cong\angle3\), statement 4).

Answer:

D. \(3.\angle3\) is supplementary to \(\angle2\), \(3\). linear pair theorem