Question

3
assume the two lines $overline{ab}$ and $overline{xy}$ intersect as in the diagram below. which of the following statements are true?
check all that apply
$angle{ary}$ and $angle{xrb}$ are vertical angles
$angle{ary}$ and $angle{xrb}$ are complementary
$overline{ab}$ and $overline{xy}$ are perpendicular
$angle{ary}$ and $angle{xrb}$ are supplementary

Explanation:

Step1: Recall vertical - angle definition

Vertical angles are formed when two lines intersect. $\angle ARY$ and $\angle XRB$ are vertical angles because they are opposite each other when lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{XY}$ intersect.

Step2: Recall complementary - angle definition

Complementary angles add up to 90 degrees. $\angle ARY$ and $\angle XRB$ are not complementary as they are vertical angles and not necessarily 45 - degree angles each.

Step3: Recall perpendicular - line definition

A line is perpendicular to another if the angle between them is 90 degrees. $\overleftrightarrow{AB}$ and $\overleftrightarrow{XY}$ are perpendicular, but it's not correct to say a single - point (like $B$) and a line ($\overleftrightarrow{XY}$) are perpendicular.

Step4: Recall supplementary - angle definition

Supplementary angles add up to 180 degrees. $\angle ARY$ and $\angle XRB$ are not supplementary as they are vertical angles and equal in measure, not adding up to 180 degrees.

Answer:

$\angle ARY$ and $\angle XRB$ are vertical angles.