Question

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

a. corresponding angles converse
b. alternate interior angles converse
c. alternate exterior angles converse
d. consecutive interior angles converse
e. consecutive exterior angles converse
f. no lines are parallel.

Explanation:

Step1: Analyze \(\angle8\cong\angle19\)

\(\angle8\) and \(\angle19\) are alternate interior angles. If alternate interior angles are congruent, then the lines are parallel. So, \(l\parallel n\) by Alternate Interior Angles Converse.

Step2: Analyze \(\angle13\cong\angle15\)

\(\angle13\) and \(\angle15\) are corresponding angles. If corresponding angles are congruent, then the lines are parallel. So, \(j\parallel k\) by Corresponding Angles Converse.

Step3: Analyze \(m\angle9 + m\angle21=180\)

\(\angle9\) and \(\angle21\) are consecutive exterior angles. There is no converse theorem for consecutive exterior angles to prove parallel lines. So, no lines are parallel.

Step4: Analyze \(\angle4\cong\angle23\)

\(\angle4\) and \(\angle23\) are alternate exterior angles. If alternate exterior angles are congruent, then the lines are parallel. So, \(l\parallel n\) by Alternate Exterior Angles Converse.

Step5: Analyze \(m\angle6 + m\angle19 = 180\)

\(\angle6\) and \(\angle19\) are not in a standard angle - relationship (like corresponding, alternate interior/exterior, consecutive interior) that can be used to prove parallel lines. So, no lines are parallel.

Step6: Analyze \(m\angle14 + m\angle15=180\)

\(\angle14\) and \(\angle15\) are consecutive interior angles. If consecutive interior angles are supplementary, then the lines are parallel. So, \(j\parallel k\) by Consecutive Interior Angles Converse.

Answer:

Angle Relationship	Parallel Lines	Converse
B. \(\angle13\cong\angle15\)	\(j\parallel k\)	A
C. \(m\angle9 + m\angle21 = 180\)	No lines are parallel	-
D. \(\angle4\cong\angle23\)	\(l\parallel n\)	C
E. \(m\angle6 + m\angle19=180\)	No lines are parallel	-
F. \(m\angle14 + m\angle15 = 180\)	\(j\parallel k\)	D