Question

assume that the lines (overline{de}) and (overline{xy}) intersect as in the diagram below. which of the following statements are true? check all that apply. a. (angle xae) and (angle day) are complementary. b. (overline{de}) and (overline{xy}) are perpendicular. c. (angle xae) and (angle eay) form a linear pair. d. (angle xae) and (angle day) are vertical angles.

Explanation:

Step1: Recall angle - pair definitions

Complementary angles add up to 90°. Vertical angles are opposite each other when two lines intersect. A linear - pair of angles are adjacent and their non - common sides form a straight line. Perpendicular lines intersect at a 90° angle.

Step2: Analyze option A

Since \(\angle XAE+\angle DAE = 90^{\circ}\) and \(\angle DAE=\angle DAY\) (vertical angles), \(\angle XAE+\angle DAY = 90^{\circ}\), so \(\angle XAE\) and \(\angle DAY\) are complementary.

Step3: Analyze option B

The right - angle symbol at the intersection of \(\overleftrightarrow{DE}\) and \(\overleftrightarrow{XY}\) indicates that they are perpendicular.

Step4: Analyze option C

\(\angle XAE\) and \(\angle EAY\) are adjacent and their non - common sides \(\overrightarrow{AX}\) and \(\overrightarrow{AY}\) form a straight line, so they form a linear pair.

Step5: Analyze option D

\(\angle XAE\) and \(\angle DAY\) are not vertical angles. Vertical angles are formed by two intersecting lines and are opposite each other. Here, \(\angle XAD\) and \(\angle EAY\) are vertical angles, and \(\angle XAE\) and \(\angle DAY\) are not.

Answer:

A. \(\angle XAE\) and \(\angle DAY\) are complementary.
B. \(\overleftrightarrow{DE}\) and \(\overleftrightarrow{XY}\) are perpendicular.
C. \(\angle XAE\) and \(\angle EAY\) form a linear pair.