Converse justifies the answer for cases (c), (d), and (f).
Based on the diagram and the given angle relationships, here's which lines, if any, can be proven parallel and the converse to justify the answer:
a. <8 ≅ <19
No lines can be proven parallel based on this angle relationship alone.
Converse: Corresponding Angles Converse: If two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel.
b. <13 ≅ <15
No lines can be proven parallel based on this angle relationship alone.
Converse: Corresponding Angles Converse: If two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel.
c. m<9 + m<21 = 180°
Lines 9 and 21 can be proven parallel based on this angle relationship. Since the sum of the measures of angles 9 and 21 is 180°, they are supplementary angles. Supplementary angles formed by a transversal cutting parallel lines are congruent. Therefore, lines 9 and 21 are parallel.
Converse: Consecutive Interior Angles Converse: If two lines are cut by a transversal and the consecutive interior angles on the same side of the transversal are supplementary, then the lines are parallel.
d. m<6 + m<19 = 180°
Lines 6 and 19 can be proven parallel based on this angle relationship. Similar to case (c), the sum of angles 6 and 19 being 180° makes them supplementary angles, and since supplementary angles formed by a transversal cutting parallel lines are congruent, lines 6 and 19 are parallel.
Converse: Consecutive Interior Angles Converse: If two lines are cut by a transversal and the consecutive interior angles on the same side of the transversal are supplementary, then the lines are parallel.
e. <4 ≅<23
No lines can be proven parallel based on this angle relationship alone. We need more information, such as the relationship between lines 4 and 23, to determine if they are parallel.
Converse: Corresponding Angles Converse: If two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel.
f. m<14 + m<15 = 180°
Lines 14 and 15 can be proven parallel based on this angle relationship. Similar to cases (c) and (d), the sum of angles 14 and 15 being 180° makes them supplementary angles, and since supplementary angles formed by a transversal cutting parallel lines are congruent, lines 14 and 15 are parallel.
Converse: Consecutive Interior Angles Converse: If two lines are cut by a transversal and the consecutive interior angles on the same side of the transversal are supplementary, then the lines are parallel.
In summary, lines 9 and 21, 6 and 19, and 14 and 15 can be proven parallel based on the given angle relationships, while lines 8 and 19, 13 and 15, and 4 and 23 cannot be proven parallel without more information. The Consecutive Interior Angles Converse justifies the answer for cases (c), (d), and (f).