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Because the half-plane metric is a conformal rescaling of the Euclidean metric, its angles are Euclidean angles. For each $y>0$, the hyperbolic lines meeting the imaginary axis at a fixed angle $\alpha$ at $iy$ are the Euclidean circleswhere $Y$ denotes the vertical coordinate. These circles have centres on the real axis and are therefore hyperbolic lines; varying $y$ gives the required collection.

To construct a triangle, put one vertex at $i$ and a second at $it$ on ${\ell }_{+}$, with $t>1$. Draw from these vertices the lines making interior angles $\alpha$ and $\beta$ with ${\ell }_{+}$. Their Euclidean centres and radii areIf their other intersection has angle $\gamma$, the law of cosines in the Euclidean triangle formed by the two centres and that intersection givesEquivalently,The right side is greater than one exactly when $\alpha +\beta +\gamma <\pi$. Since $(t+t^{-1})/2$ ranges continuously and strictly increasingly from $1$ to infinity for $t>1$, such a $t$ exists. The three lines therefore bound a hyperbolic triangle with the prescribed angles.