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, 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 perpendicular transversal p
e) figure with lines y, z and transversal p, angles 155°
f) figure with lines h, k and triangle at c, x=y+z
f) are h and k parallel lines? why or why not?
a. no, because adjacent angles are complementary.
b. yes, because a triangle fits between h and k.
c. yes, because alternate interior angles are congruent.
d. no, because adjacent angles are supplementary.

Explanation:

To determine if \( h \) and \( k \) are parallel, we use the converse of the alternate - interior angles theorem. The theorem states that if alternate interior angles are congruent, then the two lines cut by a transversal are parallel. In the figure, we know that \( x=y + z \). If we consider the angles formed by the transversal and the lines \( h \) and \( k \), and the triangle in between, we can see that the alternate interior angles (related to the angles \( y \), \( z \) and \( x \)) satisfy the condition for the converse of the alternate - interior angles theorem.

Option A is incorrect because there is no indication that adjacent angles are complementary. Option B is incorrect as the fact that a triangle fits between them is not a valid reason for parallelism. Option D is incorrect as adjacent angles being supplementary is not the relevant condition here. Option C is correct because when we analyze the angles (using the given \( x = y+z \) and the properties of angles in a triangle and alternate interior angles), we find that the alternate interior angles are congruent, which implies that \( h \) and \( k \) are parallel.

Answer:

C. Yes, because alternate interior angles are congruent