Question

lines m and n are cut by transversal l. which angle relationships are correct? check all that apply. ∠1 and ∠8 are alternate exterior angles. ∠4 and ∠6 are same side interior angles. ∠5 and ∠7 are vertical angles. ∠2 and ∠8 are corresponding angles. ∠3 and ∠6 are alternate interior angles.

Explanation:

Step1: Recall angle - relationship definitions

• Alternate exterior angles: Two angles that lie outside the two lines and on opposite sides of the transversal.
• Same - side interior angles: Two angles that lie between the two lines and on the same side of the transversal.
• Vertical angles: Two non - adjacent angles formed by the intersection of two lines.
• Corresponding angles: Two angles that are in the same relative position with respect to the lines and the transversal.
• Alternate interior angles: Two angles that lie between the two lines and on opposite sides of the transversal.

Step2: Analyze each option

• For \(\angle1\) and \(\angle8\):
• \(\angle1\) and \(\angle8\) lie outside the lines \(m\) and \(n\) and on opposite sides of the transversal \(l\). So, \(\angle1\) and \(\angle8\) are alternate exterior angles.
• For \(\angle4\) and \(\angle6\):
• \(\angle4\) and \(\angle6\) lie between the lines \(m\) and \(n\) and on the same side of the transversal \(l\). So, \(\angle4\) and \(\angle6\) are same - side interior angles.
• For \(\angle5\) and \(\angle7\):
• \(\angle5\) and \(\angle7\) are non - adjacent angles formed by the intersection of two lines (the transversal \(l\) and line \(n\)). So, \(\angle5\) and \(\angle7\) are vertical angles.
• For \(\angle2\) and \(\angle8\):
• \(\angle2\) and \(\angle8\) are not in the same relative position. \(\angle2\) and \(\angle6\) are corresponding angles.
• For \(\angle3\) and \(\angle6\):
• \(\angle3\) and \(\angle6\) are not alternate interior angles. \(\angle3\) and \(\angle6\) are not in the correct position (alternate interior angles lie between the two lines and on opposite sides of the transversal). \(\angle4\) and \(\angle6\) are same - side interior angles, \(\angle3\) and \(\angle5\) are alternate interior angles.

Answer:

\(\angle1\) and \(\angle8\) are alternate exterior angles, \(\angle4\) and \(\angle6\) are same side interior angles, \(\angle5\) and \(\angle7\) are vertical angles. So the correct options are: \(\angle1\) and \(\angle8\) are alternate exterior angles, \(\angle4\) and \(\angle6\) are same side interior angles, \(\angle5\) and \(\angle7\) are vertical angles.