Question

use the figures to determine whether the indicated lines are parallel. indicate why or why not.
a) figure with lines m, n and transversal p, angles 144° and 36°
b) figure with lines q, r and transversal p, angles 97° and 83°
c) figure with lines s, t and triangle abc, angles 75° at a and c
d) figure with lines u, v and transversal p, right angles
e) figure with lines y, z and transversal p, angles 155°
f) figure with lines h, k and point c, angles y, x, z with x=y+z
...
c) are s and t parallel lines? why or why not?
a. no, because a triangle fits between s and t
b. yes, because adjacent angles are 75° each
c. yes, because alternate interior angles are 75° each.
d. no, because corresponding angles are 105° each.

Explanation:

To determine if lines \( s \) and \( t \) are parallel, we use the Alternate Interior Angles Theorem, which states that if alternate interior angles are congruent, then the lines cut by a transversal are parallel. In the figure, the alternate interior angles formed by transversal \( AC \) (or \( AB \)) with lines \( s \) and \( t \) are both \( 75^\circ \).

• Option A is incorrect because the presence of a triangle does not determine parallelism.
• Option B is incorrect as "adjacent angles" is not the relevant criterion for parallel lines here.
• Option D is incorrect because the corresponding angles (if we consider other angles) would not be \( 105^\circ \) in a way that disproves parallelism, and the key here is the alternate interior angles of \( 75^\circ \) being congruent.
• Option C correctly identifies that the alternate interior angles are \( 75^\circ \) each, so by the Alternate Interior Angles Theorem, \( s \) and \( t \) are parallel.

Answer:

C. Yes, because alternate interior angles are \( 75^\circ \) each.