Question

two lines are intersected by a third line.
if $angle 2 cong angle 6$, which must be true about $angle 2$
$\bigcirc$ $angle 2 cong angle 5$
$\bigcirc$ $angle 2$ is complementary to $angle 5$.
$\bigcirc$ $mangle 2 = mangle 8$
$\bigcirc$ $angle 2$ is supplementary to $angle 8$.

Explanation:

Step1: Identify parallel lines condition

If $\angle 2 \cong \angle 6$, the two horizontal lines are parallel (corresponding angles converse theorem).

Step2: Analyze $\angle 2$ and $\angle 8$

$\angle 6$ and $\angle 8$ are supplementary (linear pair), so $m\angle 6 + m\angle 8 = 180^\circ$. Since $\angle 2 \cong \angle 6$, substitute $\angle 2$ for $\angle 6$: $m\angle 2 + m\angle 8 = 180^\circ$.

Step3: Eliminate other options

• $\angle 2 \cong \angle 5$: $\angle 5$ is supplementary to $\angle 6$, so $\angle 2$ is supplementary to $\angle 5$, not congruent.
• $\angle 2$ complementary to $\angle 5$: Complementary sums to $90^\circ$, but they sum to $180^\circ$, so false.
• $m\angle 2 = m\angle 8$: $\angle 8$ is supplementary to $\angle 6 \cong \angle 2$, so their measures are not equal.

Answer:

$\boldsymbol{\angle 2}$ is supplementary to $\boldsymbol{\angle 8}$.