Question

the diagram below for #14.

1. determine which lines, if any, can be proved parallel given the angle relationship. give the converse to justify your answer.

corresponding angles converse
alternate interior angles converse
alternate exterior angles converse
consecutive interior angles converse
consecutive exterior angles converse

Explanation:

Step1: Recall angle - parallel line theorems

• Corresponding Angles Converse: If two lines are cut by a transversal and the corresponding angles are congruent, then the two lines are parallel.
• Alternate Interior Angles Converse: If two lines are cut by a transversal and the alternate interior angles are congruent, then the two lines are parallel.
• Alternate Exterior Angles Converse: If two lines are cut by a transversal and the alternate exterior angles are congruent, then the two lines are parallel.
• Consecutive Interior Angles Converse: If two lines are cut by a transversal and the consecutive interior angles are supplementary, then the two lines are parallel.
• Consecutive Exterior Angles Converse: If two lines are cut by a transversal and the consecutive exterior angles are supplementary, then the two lines are parallel.

Step2: Analyze each angle relationship

• a. \(\angle8\cong\angle19\)
• \(\angle8\) and \(\angle19\) are alternate exterior angles. By the Alternate Exterior Angles Converse, if \(\angle8\cong\angle19\), then \(l\parallel n\).
• b. \(\angle13\cong\angle15\)
• \(\angle13\) and \(\angle15\) are alternate interior angles. By the Alternate Interior Angles Converse, if \(\angle13\cong\angle15\), then \(j\parallel k\).
• c. \(m\angle9 + m\angle21=180^{\circ}\)
• \(\angle9\) and \(\angle21\) are consecutive exterior angles. By the Consecutive Exterior Angles Converse, if \(m\angle9 + m\angle21 = 180^{\circ}\), then \(j\parallel k\).
• d. \(m\angle6 + m\angle19=180^{\circ}\)
• There is no direct theorem (corresponding, alternate - interior/exterior, consecutive - interior/exterior) that can be used to prove two lines parallel from this angle - sum relationship.
• e. \(\angle4\cong\angle23\)
• \(\angle4\) and \(\angle23\) are corresponding angles. By the Corresponding Angles Converse, if \(\angle4\cong\angle23\), then \(l\parallel n\).
• f. \(m\angle14 + m\angle15=180^{\circ}\)
• \(\angle14\) and \(\angle15\) are consecutive interior angles. By the Consecutive Interior Angles Converse, if \(m\angle14 + m\angle15=180^{\circ}\), then \(j\parallel k\).

Answer:

Angle Relationship	Parallel Lines	Converse
b. \(\angle13\cong\angle15\)	\(j\parallel k\)	Alternate Interior Angles Converse
c. \(m\angle9 + m\angle21 = 180^{\circ}\)	\(j\parallel k\)	Consecutive Exterior Angles Converse
d. \(m\angle6 + m\angle19=180^{\circ}\)	-	-
e. \(\angle4\cong\angle23\)	\(l\parallel n\)	Corresponding Angles Converse
f. \(m\angle14 + m\angle15 = 180^{\circ}\)	\(j\parallel k\)	Consecutive Interior Angles Converse