Answer:
Explanation:
the regular pentagon ABCDE rotates counterclockwise about its center to form pentagon A'B'C'D'E'. The angle of rotation at which point A' coincides with point D is 72 degrees.
To understand why the angle of rotation is 72 degrees, let's look at the properties of a regular pentagon. A regular pentagon has five sides of equal length and five angles of equal measure. In this case, the angle measure is 108 degrees (since the sum of the interior angles of a pentagon is 540 degrees, and there are 5 angles).
When the regular pentagon ABCDE rotates counterclockwise, each vertex moves to a new position. In this case, we are interested in the position of vertex A when it coincides with vertex D. Since both A and D are adjacent vertices in the original pentagon, the angle of rotation needed for A to coincide with D is the same as the angle measure of each interior angle of the pentagon.
Therefore, the angle of rotation is 108 degrees. However, since we want the angle between A and D, we divide the total angle of rotation by 2, giving us 108/2 = 54 degrees.
But we need to find the angle of rotation that brings A to D, which is the exterior angle of the pentagon. The exterior angle is supplementary to the interior angle, so it is equal to 180 - 54 = 126 degrees.
However, since we want the angle of rotation at which A' (the new position of A) coincides with D, we subtract the interior angle measure of the pentagon (108 degrees) from the exterior angle (126 degrees), giving us 126 - 108 = 18 degrees.
Therefore, the angle of rotation at which point A' coincides with point D is 18 degrees.