The tangents on the end points of $a$ diameter of $a$ circle are always ________.

Step 1 (Setup): Let $AB$ be a diameter of a circle with center $O$. Let line $L_1$ be the tangent at endpoint $A$ and line $L_2$ be the tangent at endpoint $B$.

Step 2 (Perpendicularity of Tangents): By the radius-tangent perpendicularity theorem:

• The tangent $L_1$ at $A$ is perpendicular to the radius $OA$ (and hence perpendicular to the diameter $AB$). Thus, $\angle L_{1}AB=90^{\circ }$.

• The tangent $L_2$ at $B$ is perpendicular to the radius $OB$ (and hence perpendicular to the diameter $AB$). Thus, $\angle L_{2}BA=90^{\circ }$.

Step 3 (Parallel Lines Condition): Diameter $AB$ acts as a transversal line cutting across lines $L_1$ and $L_2$. Since both consecutive interior angles sum to $180^{\circ }$ (or alternate interior angles are equal to $90^{\circ }$):

Lines $L_1$ and $L_2$ must be parallel to each other.

Step 4 (Conclusion): The tangents are always parallel.