Question

1. given: ∠2 and ∠7 are supplementary. prove: m || n. 18. given: lines m and t intersect. prove: ∠3 and ∠4 are supplementary. a. definition of supplementary angles. b. if two lines form congruent alternate interior angles with a transversal, then the lines are parallel. c. if two lines form congruent corresponding angles, then the lines are parallel. d. if two lines form supplementary interior angles on the same side of a transversal, then the lines are parallel. e. if two lines form congruent alternate exterior angles, then the lines are parallel.

Explanation:

Step1: Recall angle - line relationships

When two parallel lines are cut by a transversal, there are specific angle - angle relationships such as corresponding, alternate interior/exterior, and same - side interior/exterior angles. Supplementary angles add up to 180 degrees.

Step2: Analyze the given statements

For the first problem, given \(m\parallel n\) and \(\angle2\) and \(\angle7\) are supplementary. We know that when two parallel lines are cut by a transversal, same - side interior angles are supplementary. Here \(\angle2\) and \(\angle7\) are same - side interior angles.
For the second problem, when lines \(m\) and \(t\) intersect, we use the linear - pair postulate. A linear pair of angles are supplementary. If we assume \(\angle3\) and \(\angle4\) form a linear pair (since they are adjacent and their non - common sides are opposite rays when \(m\) and \(t\) intersect), then by the definition of a linear pair, \(\angle3+\angle4 = 180^{\circ}\), so they are supplementary.

Step3: Prove the parallel - line statements

For the statements about parallel lines:

• If two lines form congruent alternate interior angles with a transversal, then the lines are parallel (alternate interior angles converse).
• If two lines form congruent corresponding angles, then the lines are parallel (corresponding angles converse).
• If two lines form congruent alternate exterior angles, then the lines are parallel (alternate exterior angles converse).
• If two lines form supplementary interior angles on the same side of a transversal, then the lines are parallel (same - side interior angles converse).

Answer:

1. Given \(m\parallel n\), \(\angle2\) and \(\angle7\) are same - side interior angles. By the same - side interior angles postulate for parallel lines, they are supplementary.
2. When lines \(m\) and \(t\) intersect, \(\angle3\) and \(\angle4\) form a linear pair. By the definition of a linear pair, \(\angle3\) and \(\angle4\) are supplementary.
3. The statements about parallel lines are based on the converse of angle - line relationships:

• Converse of alternate interior angles theorem: If two lines form congruent alternate interior angles with a transversal, the lines are parallel.
• Converse of corresponding angles theorem: If two lines form congruent corresponding angles, the lines are parallel.
• Converse of alternate exterior angles theorem: If two lines form congruent alternate exterior angles, the lines are parallel.
• Converse of same - side interior angles theorem: If two lines form supplementary interior angles on the same side of a transversal, the lines are parallel.