Question

which is enough information to prove that $s\parallel t$?$\angle 1 \cong \angle 3$$\angle 2 \cong \angle 8$$\angle 5+\angle 6 = 180^\circ$$\angle 1+\angle 4 = 180^\circ$

Explanation:

Step1: Analyze each option

Option 1: $\angle 1 \cong \angle 3$

These are vertical angles, which are always congruent regardless of whether $s$ and $t$ are parallel. This does not prove $s \parallel t$.

Option 2: $\angle 2 \cong \angle 8$

$\angle 2$ and $\angle 8$ are alternate exterior angles. By the converse of the alternate exterior angles theorem, if alternate exterior angles are congruent, then the two lines cut by the transversal are parallel. This can prove $s \parallel t$.

Option 3: $\angle 5 + \angle 6 = 180^\circ$

These are supplementary adjacent angles (a linear pair), which are always supplementary regardless of parallelism. This does not prove $s \parallel t$.

Option 4: $\angle 1 + \angle 4 = 180^\circ$

These are supplementary adjacent angles (a linear pair), which are always supplementary regardless of parallelism. This does not prove $s \parallel t$.

Answer:

$\boldsymbol{\angle 2 \cong \angle 8}$