Question

in part (a), prove that h || k. in parts (b) and (c), solve for x given that q || p and s || r, respectively
a. why is h || k?
a. alternate interior angles are congruent.
b. since x + y = 180° and if x is the corresponding angle to x, then x is supplementary to y and x + y = 180°. therefore x and x both have the same measure, 180 - y
c. since x + y = 180° and if x is the corresponding angle to x, then x is complementary to y and x + y = 180°. therefore x and x both have the same measure, 90 - y
d. since x and y are supplementary angles, the lines are parallel

Explanation:

To determine why \( h \parallel k \), we analyze the options:

• Option A: Alternate interior angles being congruent is for parallel lines, but here we have \( x + y = 180^\circ \), not congruent alternate interior angles. Eliminate A.
• Option B: If \( x' \) is the corresponding angle to \( x \), and \( x + y = 180^\circ \), then \( x' + y = 180^\circ \) (since corresponding angles would have the same relationship with \( y \)). So \( x' = 180^\circ - y \) and \( x = 180^\circ - y \), meaning \( x' = x \). Corresponding angles being equal implies lines are parallel. This makes sense.
• Option C: Complementary angles sum to \( 90^\circ \), but \( x + y = 180^\circ \) (supplementary), so this is incorrect. Eliminate C.
• Option D: Supplementary angles on their own don't imply parallel lines; the relationship with corresponding angles (as in Option B) is needed. Eliminate D.

Answer:

B. Since \( x + y = 180^\circ \) and if \( x' \) is the corresponding angle to \( x \), then \( x' \) is supplementary to \( y \) and \( x' + y = 180^\circ \). Therefore \( x' \) and \( x \) both have the same measure, \( 180 - y \)