On a plane, a line $\ell$ and two circles $c_1$ and $c_2$ of different radii are given such that  $\ell$  touches both circles at point $P$. Point $M \ne P$ on $\ell$ is chosen so that the angle $Q_1MQ_2$ is as large as possible where $Q_1$ and $Q_2$ are the tangency points of the tangent lines drawn from $M$ to $c_i$ and $c_2$, respectively, differing from $\ell$ . Find $\angle PMQ_1 + \angle PMQ_2$·
1. **Understanding the Problem:**
   - We have a line $\ell$ that is tangent to two circles $c_1$ and $c_2$ at point $P$.
   - We need to find a point $M \ne P$ on $\ell$ such that the angle $\angle Q_1MQ_2$ is maximized, where $Q_1$ and $Q_2$ are the points of tangency of the tangent lines drawn from $M$ to $c_1$ and $c_2$, respectively.

2. **Analyzing the Geometry:**
   - Since $\ell$ is tangent to both circles at $P$, $P$ is the point of tangency for both circles.
   - The tangent lines from $M$ to $c_1$ and $c_2$ will touch the circles at $Q_1$ and $Q_2$, respectively.

3. **Maximizing the Angle $\angle Q_1MQ_2$:**
   - To maximize $\angle Q_1MQ_2$, $M$ should be chosen such that the tangents $MQ_1$ and $MQ_2$ are as far apart as possible.
   - The maximum angle between two lines is $180^\circ$.

4. **Considering the Configuration:**
   - When $M$ is chosen such that $\angle Q_1MQ_2 = 180^\circ$, $Q_1$ and $Q_2$ must lie on a straight line passing through $M$.
   - This implies that $M$ lies on the line $\ell$ such that $Q_1$ and $Q_2$ are collinear with $M$.

5. **Summing the Angles:**
   - Since $\angle Q_1MQ_2 = 180^\circ$, the sum of the angles $\angle PMQ_1$ and $\angle PMQ_2$ must also be $180^\circ$ because they are supplementary angles formed by the tangents and the line $\ell$.

Therefore, the sum of the angles $\angle PMQ_1$ and $\angle PMQ_2$ is $180^\circ$.

The final answer is $\boxed{180^\circ}$