Question

choose the theorem that would best be used to prove the following statement: given that lines a and b are parallel. if the m∠3 = 115°, then m∠5 = 65°
diagram: two parallel lines (a, b) cut by a transversal, with angles labeled 1,2,3,4 (on line a) and 5,6,7,8 (on line b)
options:

• alternate interior angles theorem
• alternate exterior angles theorem
• consecutive interior angles theorem
• vertical angles theorem

Explanation:

1. Recall the definitions of each theorem:

• Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
• Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then alternate exterior angles are congruent.
• Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal, then consecutive interior angles are supplementary (their sum is \(180^\circ\)).
• Vertical Angles Theorem: Vertical angles are congruent.

1. Analyze the angles \(\angle 3\) and \(\angle 5\):

• \(\angle 3\) and \(\angle 5\) are consecutive interior angles (they lie between the two parallel lines \(a\) and \(b\) and on the same side of the transversal).
• Given \(m\angle 3 = 115^\circ\) and we need to find \(m\angle 5 = 65^\circ\). Notice that \(115^\circ+ 65^\circ=180^\circ\), which means they are supplementary.
• The Consecutive Interior Angles Theorem states that consecutive interior angles formed by a transversal cutting two parallel lines are supplementary. This matches the relationship between \(\angle 3\) and \(\angle 5\).
• The Alternate Interior Angles Theorem would apply to angles like \(\angle 3\) and \(\angle 6\) (or \(\angle 4\) and \(\angle 5\)), the Alternate Exterior Angles Theorem to angles like \(\angle 1\) and \(\angle 8\) (or \(\angle 2\) and \(\angle 7\)), and the Vertical Angles Theorem to angles like \(\angle 1\) and \(\angle 3\) (or \(\angle 2\) and \(\angle 4\), etc.), so they do not apply here.

Answer:

Consecutive Interior Angles Theorem