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  A,B,C,  and D are distinct points on the plane. A rotation about point O maps  A to B, B  to C, and C to D. We know angle AOD is 24 Enter all three possible degree measures (between 0 and 180) of angle AOB separated by commas.

 Apr 13, 2024
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Since a rotation about point O maps A to B, B to C, and C to D, we know the following:

 

Angle AOD: This is given as 24 degrees.

 

Full Rotation: A full rotation around a point is 360 degrees.

 

Possible Rotations:

 

There are three possibilities for the rotation that maps A to B, B to C, and C to D, resulting in three possible measures for angle AOB:

 

Case 1: Single Rotation of 24 Degrees

 

In this case, the rotation about O maps A to B by 24 degrees clockwise, B to C by 24 degrees clockwise, and C to D by 24 degrees

clockwise, resulting in a total rotation of A to D of 24 + 24 + 24 = 72 degrees.

 

Since the full rotation is 360 degrees, the remaining angle for AOB must be 360 - 72 = 288 degrees.

 

However, angle measures are typically represented between 0 and 180 degrees. We can achieve this by subtracting a multiple of 180 (full rotations) from 288:

 

AOB = 288° - 180 = 108.

 

Case 2: Double Rotation (360 + 24 Degrees)

 

Here, the rotation about O maps A to B by a full rotation (360 degrees) followed by a 24-degree clockwise rotation. This effectively brings B back to its original position and then continues the rotation to C and D.

 

The total rotation for AOD in this case becomes 360 + 24 + 24 = 408 degrees.

 

Similar to case 1, we can adjust this to fit the 0-180 degree range:

 

AOB = 408° - 360° = 48°

 

Case 3: Triple Rotation (2 * 360 + 24 Degrees)

 

This case involves two full rotations followed by a 24-degree clockwise rotation from A to B. Again, the first two rotations effectively bring B back to its original position.

 

The total rotation for AOD becomes 2 * 360 + 24 = 744 degrees.

 

Adjusting for the 0-180 degree range:

 

AOB = 744° - 2 * 360° = 124°

 

Summary:

 

Therefore, the three possible degree measures for angle AOB, considering rotations between 0 and 180 degrees, are:

 

108° (Case 1)

 

48° (Case 2)

 

124° (Case 3)

 Apr 14, 2024

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