Question

consider the drawing shown.

image of intersecting lines with angles 1, 2, 3

which of the following information would be sufficient to demonstrate two lines are parallel?

a. ∠1 is congruent to ∠2 and supplementary to ∠3.

b. ∠1 is supplementary to ∠2 and congruent to ∠3.

c. ∠1 is congruent to both ∠2 and ∠3.

d. ∠1 is supplementary to both ∠2 and ∠3.

Explanation:

To solve this problem, we analyze each option using properties of parallel lines (corresponding angles, alternate interior angles, supplementary angles):

Option Analysis:

• Option a: $\angle 1 \cong \angle 2$ (suggests parallelism, but $\angle 1$ supplementary to $\angle 3$ does not confirm $\angle 2 \parallel \angle 3$ logic).
• Option b: $\angle 1$ supplementary to $\angle 2$ (linear pair, always true) and $\angle 1 \cong \angle 3$ (alternate interior, suggests parallelism, but not enough for $a \parallel b$).
• Option c: $\angle 1 \cong \angle 2$ (corresponding angles, implies $a \parallel b$) and $\angle 1 \cong \angle 3$ (alternate interior, implies $c \parallel d$). This satisfies the need to show $a \parallel b$ and $c \parallel d$.
• Option d: $\angle 1$ supplementary to both $\angle 2$ and $\angle 3$ (linear pair is always supplementary, but congruence is needed for parallelism, not supplementary).

Answer:

c. $\angle 1$ is congruent to both $\angle 2$ and $\angle 3$