geometry constructions In hyperbolic geometry, constructions that can be done using any one of the three compasses listed above and the parallel ruler can also be done using the hyperbolic ruler Simple constructions Angle bisector Consider a given angle ᗉ IAI' ≠ /2 radians whose angle bisector is sought. This results in two different cases: either ᗉ IAI' < /2 radians or ᗉ IAI' > /2 radians. For both cases a hyperbolic ruler is needed to construct a line BI' where BI' is perpendicular to AI and parallel to AI'. Also, construct a line B'I where B'I is perpendicular to AI' and parallel to AI. Case 1: ᗉ IAI'< /2 radians Let C be the intersection of BI' and B'I. The result of this is that the line AC bisects ᗉ IAI'. Case 2: ᗉ IAI' > /2 radians This case is further broken down into three sub-cases: Case 2a: IB' intersects I'B Let A' be the intersection of IB' and I'B. Then AA' is the angle bisector of ᗉ IAI'. Case 2b: IB' is parallel to I'B Construct the line segment BB' and using a hyperbolic ruler, construct the line OI" such that OI" is perpendicular to BB' and parallel to B'I". Then, line OA is the angle bisector for ᗉ IAI'. Case 2c: IB' is ultraparallel to I'B. Using the ultraparallel theorem, construct the common perpendicular of IB' and I'B, CC'. Let the intersection of CB" and BC' be D. As a result, AD will be the angle bisector of ᗉ BDB'. We then find that the line through OD is also the angle bisector of ᗉ IAI'. Common parallel line to two lines We consider the problem of finding a line parallel to two given lines, a and a'. There are three cases: a and a' intersect at a point O, a and a' are parallel to each other, and a and a are ultraparallel to each other. Case 1: a and a' intersect at a point O, Bisect one of the angles made by these two lines and name the angle bisector b. Using a hyperbolic ruler, construct a line c such that c is perpendicular to b and parallel to a. As a result, c is also parallel to a', making c the common parallel to lines a and a'. Case 2: a and a' are parallel to each other Using a hyperbolic ruler, construct