How to construct an angle of 105

To construct an angle of 105 degrees:

1. Draw a ray starting from a point O. Let this be OA.
2. Construct a 90-degree angle at point O. To do this, place the compass at O and draw an arc intersecting the ray OA at point P. With the same compass setting, place the compass point at P and draw an arc intersecting the first arc at Q. Again, with the same setting, place the compass point at Q and draw an arc intersecting the first arc at R. The line segment OR forms a 90-degree angle with OA.
3. Bisect the 90-degree angle. Place the compass point at R and draw an arc. Then, place the compass point at Q and draw another arc that intersects the previous arc at S. The ray OS bisects the angle ROA.
4. The angle ROS is 45 degrees. The angle QOS is also 45 degrees.
5. Add 60 degrees to 45 degrees. We know that an angle of 60 degrees can be constructed by placing the compass at P (where the first arc intersected OA) and drawing an arc that intersects the first arc at T. The ray OT forms a 60-degree angle with OA.
6. Combine angles. The angle TOA is 60 degrees. The angle QOS is 45 degrees. The angle QOT is the difference between TOA and QOA (which is 90 degrees), so QOT = 90 - 60 = 30 degrees.
7. The desired angle is 105 degrees. This can be formed by adding the 60-degree angle (TOA) and half of the 90-degree angle (which is 45 degrees). Specifically, the angle formed by ray OT and ray OS is 60 degrees + 45 degrees = 105 degrees.