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 $\angle N$. $$ \angle TOM = 2 \times \angle N \