Question

According to Theorem 2-13, which states that two non-vertical lines are parallel if and only if their slopes are equal, which of the following if-then statements is implied?

a) If two lines are parallel, then their slopes are equal.
b) If two lines have equal slopes, then they are parallel.
c) If two lines are perpendicular, then their slopes are equal.
d) If two lines have different slopes, then they are parallel.

Answer 1

The implied if-then statements by Theorem 2-13 are 'If two lines are parallel, then their slopes are equal' and 'If two lines have equal slopes, then they are parallel'. Lines A and B with slopes -4.7 and 12.0, respectively, are not parallel as their slopes are not equal.

Step-by-step explanation:

The correct if-then statement implied by Theorem 2-13 is a) If two lines are parallel, then their slopes are equal and b) If two lines have equal slopes, then they are parallel. This theorem essentially provides both the necessary and sufficient conditions for two lines to be parallel, meaning that having equal slopes is both required for lines to be parallel and guarantees that they are parallel, as long as the lines are not vertical. This is because vertical lines have an undefined slope and cannot be compared in this way. Statement c) is incorrect as perpendicular lines have slopes that are negative reciprocals of each other, not equal. Statement d) is also incorrect because if two lines have different slopes, they will intersect at some point and thus cannot be parallel.

Regarding the specific slopes provided, since Line A has a slope of -4.7 and Line B has a slope of 12.0, we can conclude that these two lines are not parallel since their slopes are not equal.