Question

which of the following theorems about angles 4 and 6 would guarantee that lines a and b are parallel? image of two parallel lines (a and b) cut by a transversal, with angles labeled 1,2,3,4 (on line a) and 5,6,7,8 (on line b) options: the alternate interior angle converse theorem; the same - side interior angle converse theorem; the alternate exterior angle converse theorem; the corresponding angle converse theorem

Explanation:

Step1: Identify Angle Relationship

Angles 4 and 6 are same - side interior angles (they lie between the two lines \(a\) and \(b\) and on the same side of the transversal).

Step2: Recall Converse Theorems

• The alternate interior angle converse theorem applies to alternate interior angles (non - adjacent, on opposite sides of the transversal, between the lines). Angles 4 and 6 are not alternate interior angles.
• The same - side interior angle converse theorem states that if two same - side interior angles are supplementary, then the two lines are parallel. Since angles 4 and 6 are same - side interior angles, the same - side interior angle converse theorem is the one that would guarantee lines \(a\) and \(b\) are parallel if the relevant angle condition (supplementary) is met.
• The alternate exterior angle converse theorem applies to alternate exterior angles (outside the two lines, on opposite sides of the transversal), which is not the case for angles 4 and 6.
• The corresponding angle converse theorem applies to corresponding angles (in the same relative position at each intersection), which is not the case for angles 4 and 6.

Answer:

the same - side interior angle converse theorem