Identify the angle between the radius and the tangent.

Here are the calculations for the angles, with reasons:

4.1 Calculate $\angle TOM$

Step 1: Identify the angle between the radius and the tangent.
The radius $OM$ is perpendicular to the tangent $MP$ at the point of tangency $M$.

$$\angle OMP = 90^\circ \quad (\text{Radius} \perp \text{Tangent})$$

Step 2: Use the given angle $\angle PMT$ to find $\angle OMT$.
We are given $\angle PMT = 40^\circ$.

$$\angle OMT = \angle OMP - \angle PMT$$ $$\angle OMT = 90^\circ - 40^\circ$$ $$\angle OMT = 50^\circ$$

Step 3: Identify $\triangle OMT$ as an isosceles triangle.
$OT$ and $OM$ are radii of the same circle.

$$OT = OM \quad (\text{Radii})$$

Therefore, $\triangle OMT$ is an isosceles triangle.

Step 4: Find $\angle OTM$.
In isosceles $\triangle OMT$, the base angles are equal.

$$\angle OTM = \angle OMT = 50^\circ \quad (\text{Base angles of isosceles triangle})$$

Step 5: Calculate $\angle TOM$.
The sum of angles in a triangle is $180^\circ$.

$$\angle TOM + \angle OTM + \angle OMT = 180^\circ \quad (\text{Sum of angles in a triangle})$$ $$\angle TOM + 50^\circ + 50^\circ = 180^\circ$$ $$\angle TOM + 100^\circ = 180^\circ$$ $$\angle TOM = 180^\circ - 100^\circ$$ $$\angle TOM = \boxed{80^\circ}$$

4.2 Calculate $\angle N$

Step 1: Apply the Angle at Center Theorem.
The angle subtended by arc $MT$ at the center is $\angle TOM$.
The angle subtended by arc $MT$ at the circumference is $\angle N$.

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