1. **Problem statement:** We have a hyperbolic line $l$, a point $A$ on $l$, and a point $B$ outside $l$. We need to construct: (a) One hyperbolic line $h_1$ parallel to $l$ passing through $B$, and one hyperbolic line $h_2$ intersecting $l$ passing through $B$. (b) The hyperbolic line passing through both $A$ and $B$. 2. **Key concepts:** - In hyperbolic geometry, through a point not on a given line, there are infinitely many lines that do not intersect the given line (parallel lines). - A hyperbolic line is represented by an arc of a circle orthogonal to the boundary circle $H$. - To construct lines through $B$, we find arcs orthogonal to $H$ passing through $B$. 3. **Step (a) construction:** - To construct $h_1$ (parallel to $l$ through $B$): - Identify the circle defining $l$ (orthogonal to $H$). - Construct a circle orthogonal to $H$ passing through $B$ that does not intersect $l$ inside $H$. - This circle's arc inside $H$ is $h_1$. - To construct $h_2$ (intersecting $l$ through $B$): - Construct a circle orthogonal to $H$ passing through $B$ that intersects the circle defining $l$ inside $H$. - The arc inside $H$ is $h_2$. 4. **Step (b) construction:** - The hyperbolic line through $A$ and $B$ is the unique circle orthogonal to $H$ passing through both points. - Find the circle orthogonal to $H$ that passes through $A$ and $B$. - The arc inside $H$ is the hyperbolic line through $A$ and $B$. 5. **Summary:** - $h_1$: circle orthogonal to $H$ through $B$ parallel to $l$. - $h_2$: circle orthogonal to $H$ through $B$ intersecting $l$. - Line through $A$ and $B$: unique circle orthogonal to $H$ through both points. This completes the constructions as per hyperbolic geometry rules.