Question

assume the two lines $overleftrightarrow{ab}$ and $overleftrightarrow{xy}$ intersect as in the diagram below. which of the following statements are true?
check all that apply

a. $angle ary$ and $angle xrb$ are vertical angles.
b. $angle ary$ and $angle xrb$ are supplementary.
c. $overleftrightarrow{ab}$ and $overleftrightarrow{xy}$ are perpendicular.
d. $angle ary$ and $angle xrb$ are complementary.

Explanation:

Step1: Identify perpendicular lines

The diagram shows a right angle at intersection \(R\), so \( \overline{AB} \perp \overline{XY} \).

Step2: Analyze angle types

Vertical angles are opposite intersecting lines; \( \angle ARY \) and \( \angle XRB \) are not opposite. Supplementary angles sum to \(180^\circ\); complementary sum to \(90^\circ\). Since lines are perpendicular, \( \angle ARY = 90^\circ \), \( \angle XRB = 90^\circ \). Their sum is \(180^\circ\), so they are supplementary.

Step3: Verify perpendicularity

The right angle symbol confirms \( \overline{AB} \) and \( \overline{XY} \) are perpendicular.

Answer:

B. \( \angle ARY \) and \( \angle XRB \) are supplementary.
C. \( \overline{AB} \) and \( \overline{XY} \) are perpendicular.