Question

two lines are intersected by a third line
if ( angle 2congangle 6 ), which must be true about ( angle 2 )?
( angle 2congangle 5 )
( angle 2 ) is complementary to ( angle 5 )
( mangle 2 = mangle 8 )
( angle 2 ) is supplementary to ( angle 8 )

Explanation:

Step1: Recall the properties of angles formed by parallel lines and a transversal

When two lines are intersected by a transversal, if $\angle2\cong\angle6$, then the two lines are parallel (by the converse of the corresponding angles postulate).

Step2: Analyze the relationship between $\angle2$ and $\angle8$

We know that $\angle6$ and $\angle8$ are supplementary ($\angle6+\angle8 = 180^{\circ}$) because they are adjacent angles forming a linear pair. Since $\angle2\cong\angle6$ (given), we can substitute $\angle2$ for $\angle6$ in the equation $\angle6+\angle8=180^{\circ}$. So, $\angle2+\angle8 = 180^{\circ}$, which means $\angle2$ is supplementary to $\angle8$.

Step3: Analyze other options

• For $\angle2\cong\angle5$: $\angle2$ and $\angle5$ are not corresponding or alternate - interior angles in a way that would make them congruent based on the given $\angle2\cong\angle6$.
• For $\angle2$ is complementary to $\angle5$: There is no basis for a $90^{\circ}$ sum from the given information.
• For $m\angle2=m\angle8$: Since $\angle2+\angle8 = 180^{\circ}$ (from above), they are not equal (unless each is $90^{\circ}$, but there is no information to suggest that).

Answer:

$\angle2$ is supplementary to $\angle8$