Question

given: line a and line b intersect
prove: \\( \angle 1 \cong \angle 3 \\)
which statement and reason best completes the proof?
a. 2. \\( \angle 2 \\) is supplementary to \\( \angle 4 \\) 2. linear pair theorem
b. 2. \\( m \angle 1 + m \angle 4 = 180 ^ { \circ } \\) 2. definition of supplementary angles
c. 2. \\( \angle 1 \\) is supplementary to \\( \angle 2 \\) 2. linear pair theorem
d. 2. \\( m \angle 2 + m \angle 3 = 180 ^ { \circ } \\) 2. definition of supplementary angles

Explanation:

Step1: Recall the linear pair theorem

The linear pair theorem states that if two angles form a linear pair (adjacent angles whose non - common sides are opposite rays), then they are supplementary.

Step2: Analyze each option

• Option A: $\angle2$ and $\angle4$ are not a linear pair. A linear pair must be adjacent. So, this is incorrect.
• Option B: $\angle1$ and $\angle4$ are vertical angles (not a linear pair). So, this is incorrect.
• Option C: $\angle1$ and $\angle2$ form a linear pair (adjacent angles with non - common sides as opposite rays). By the linear pair theorem, $\angle1$ is supplementary to $\angle2$.
• Option D: While $m\angle2 + m\angle3=180^{\circ}$ is true (by the linear pair theorem), the reason should be the linear pair theorem (not the definition of supplementary angles for this context, since we are using the geometric relationship of a linear pair).

Answer:

C. 2. $\angle1$ is supplementary to $\angle2$ 2. linear pair theorem