The angle between the tangents at the ends of two perpendicular radii is ________.

Step 1 (Setup and Notation): Let the circle have center $O$. Let the two perpendicular radii be $OA$ and $OB$, so $\angle AOB=90^{\circ }$. Let the tangents at $A$ and $B$ intersect at a point $P$.

Step 2 (Properties of Tangents): A tangent is always perpendicular to the radius at the point of contact. Therefore:

• $\angle OAP=90^{\circ }$

• $\angle OBP=90^{\circ }$

Step 3 (Quadrilateral Angle Sum): Consider the quadrilateral $OAPB$. The sum of the interior angles of a quadrilateral is always $360^{\circ }$:

Step 4 (Conclusion): The angle between the tangents is exactly $90^{\circ }$. The numeric value is 90.