Question

by which angle must \\( \overline{ab} \\) turn about point a in the clockwise direction so that it coincides with \\( \overline{ae} \\)?
options: a. 90° b. 180° c. 270° d. 360°

Explanation:

Step1: Analyze the diagram

From the diagram, we can see that \(\angle BAE\) is a straight angle? No, wait, looking at the right angle symbol at \(A\) between the vertical and horizontal lines. Wait, actually, \(\overline{AB}\) is vertical (upwards from \(A\)) and \(\overline{AE}\) is vertical downwards from \(A\)? Wait, no, wait the diagram: \(A\) is the vertex, \(\overline{AB}\) is going up, \(\overline{AE}\) is going down? Wait, no, the right angle symbol suggests that the horizontal and vertical lines are perpendicular. Wait, maybe \(\overline{AB}\) is along the positive y - axis (up) and \(\overline{AE}\) is along the negative y - axis (down). So the angle between \(\overline{AB}\) and \(\overline{AE}\) when rotating clockwise: a full rotation is \(360^{\circ}\), but a straight angle in the clockwise direction from \(\overline{AB}\) to \(\overline{AE}\) (if they are opposite rays) would be \(180^{\circ}\)? Wait, no, wait the options: A. \(90^{\circ}\), B. \(180^{\circ}\), C. \(270^{\circ}\), D. \(360^{\circ}\). Wait, maybe I misread the diagram. Wait, the right angle symbol at \(A\) between the horizontal (like \(AD\) and \(AC\)) and vertical ( \(AB\) and \(AE\)) lines. So \(\overline{AB}\) is vertical (let's say north) and \(\overline{AE}\) is vertical south? Wait, no, maybe \(\overline{AB}\) is along the positive y - axis and \(\overline{AE}\) is along the negative y - axis. The angle of rotation clockwise from \(\overline{AB}\) to \(\overline{AE}\): if you rotate \(\overline{AB}\) clockwise, to get to \(\overline{AE}\), which is directly opposite (since they are colinear but opposite in direction), the angle is \(180^{\circ}\)? Wait, no, wait a clockwise rotation from up to down: a full circle is \(360^{\circ}\), half - circle is \(180^{\circ}\). Wait, but maybe the diagram has \(\overline{AB}\) and \(\overline{AE}\) as opposite rays, so the clockwise angle between them is \(180^{\circ}\). Wait, but let's check the options. Option B is \(180^{\circ}\). Wait, maybe I made a mistake. Wait, no, if \(\overline{AB}\) is vertical (up) and \(\overline{AE}\) is vertical (down), the clockwise rotation from \(\overline{AB}\) to \(\overline{AE}\) is \(180^{\circ}\) because you are rotating half of a full circle (\(360/2 = 180\)) in the clockwise direction.

Step2: Determine the angle of rotation

When we rotate a ray \(\overline{AB}\) clockwise about point \(A\) to coincide with \(\overline{AE}\), if \(\overline{AB}\) and \(\overline{AE}\) are opposite rays (forming a straight line in the vertical direction), the measure of the clockwise angle between them is \(180^{\circ}\). Because a straight angle (the angle between two opposite rays) is \(180^{\circ}\), and rotating clockwise from one ray to its opposite ray gives an angle of \(180^{\circ}\).

Answer:

B. \(180^{\circ}\)