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 $\angle 5$. $\bigcirc$ $m\angle 2 = m\angle 8$ $\bigcirc$ $\angle 2$ is supplementary to $\angle 8$.

Explanation:

Step1: Recall Corresponding Angles Postulate

If \( \angle 2 \cong \angle 6 \), the two lines cut by the transversal are parallel (by Corresponding Angles Postulate converse).

Step2: Analyze \( \angle 2 \) and \( \angle 8 \)

\( \angle 6 \) and \( \angle 8 \) are vertical angles, so \( \angle 6 \cong \angle 8 \). Since \( \angle 2 \cong \angle 6 \), by transitivity, \( \angle 2 \cong \angle 8 \), so \( m\angle 2 = m\angle 8 \).

Step3: Eliminate other options

• \( \angle 2 \) and \( \angle 5 \): \( \angle 2 \) and \( \angle 5 \) are same - side interior angles (if lines are parallel, they are supplementary, not congruent), so first option wrong.
• \( \angle 2 \) and \( \angle 5 \): They are supplementary (if lines parallel), not complementary, so second option wrong.
• \( \angle 2 \) and \( \angle 8 \): They are congruent, not supplementary (since \( \angle 6+\angle 8 = 180^{\circ}\) and \( \angle 2=\angle 6\), \( \angle 2+\angle 8 = 180^{\circ}\)? Wait, no, \( \angle 6\) and \( \angle 8\) are vertical angles, so \( \angle 6=\angle 8\). If \( \angle 2=\angle 6\), then \( \angle 2 = \angle 8\), so they are congruent, so the fourth option is wrong.

Answer:

\( m\angle 2 = m\angle 8 \) (the option: \( m\angle 2 = m\angle 8 \))