Question

1. given: ∠4 = ∠5. prove: m||n

statements reasons

1. ∠4≅∠5 1. given
2. ∠1≅∠4 2.
3. ∠1≅∠5 3. transitive property
4. m||n 4.

Explanation:

Step1: Identify vertical - angle relationship

$\angle1$ and $\angle4$ are vertical angles, so $\angle1\cong\angle4$ (Vertical - angles are congruent).

Step2: Use transitive property

Given $\angle4\cong\angle5$ and $\angle1\cong\angle4$, by the transitive property of congruence, $\angle1\cong\angle5$.

Step3: Apply parallel - line criterion

$\angle1$ and $\angle5$ are corresponding angles. If corresponding angles are congruent, then the lines are parallel. So $m\parallel n$ (Corresponding angles postulate).

Answer:

1. Vertical - angles are congruent; 4. Corresponding angles postulate