Let's analyze the problem step-by-step.

The diagram shows a circle with center $O$. The line $TAE$ is a tangent to the circle at point $A$. We are given that $\angle AOD = 72^\circ$. We need to calculate the following angles:

(i) $\angle ADC$

• Since $\angle AOD$ is a central angle, the angle subtended by the same arc $AD$ at the circumference is half of the central angle.
• $\angle ADC = \frac{1}{2} \angle AOD = \frac{1}{2} \times 72^\circ = 36^\circ$.

(ii) $\angle ACD$

• $\angle ACD$ is an exterior angle of triangle $\triangle ADC$ at vertex $C$.
• According to the exterior angle theorem, an exterior angle is equal to the sum of the two non-adjacent interior angles.
• $\angle ACD = \angle CAD + \angle ADC$.
• However, we need to calculate $\angle CAD$ before finding this angle. We'll come back to it.

(iii) $\angle CAD$

• Since $\triangle AOD$ is isosceles with $OA = OD$ (both are radii), $\angle OAD = \angle ODA$.
• Using the angle sum property of a triangle: $\angle AOD + \angle OAD + \angle ODA = 180^\circ$.
• $\angle OAD = \angle ODA = \frac{180^\circ - 72^\circ}{2} = 54^\circ$.
• Since $\angle OAD$ is equal to $\angle CAD$, $\angle CAD = 54^\circ$.

(iv) $\angle OEA$

• $\angle OEA$ is the angle between the radius $OA$ and the tangent $TAE$.
• According to the tangent-radius theorem, the angle between a radius and a tangent at the point of contact is $90^\circ$.
• Thus, $\angle OEA = 90^\circ$.

Summary:

1. $\angle ADC = 36^\circ$
2. $\angle ACD = 90^\circ$ (since it is the sum of $\angle CAD$ and $\angle ADC$)
3. $\angle CAD = 54^\circ$
4. $\angle OEA = 90^\circ$

Would you like more detailed explanations for any of the steps, or have any specific questions?

Related Questions:

1. How can you prove that a tangent to a circle is perpendicular to the radius at the point of contact?
2. What is the relationship between the central angle and the inscribed angle subtended by the same arc?
3. How would the solution change if $\angle AOD$ was different?
4. How can you find angles in a circle using the property of cyclic quadrilaterals?
5. Can you explain the exterior angle theorem with another example?

Tip: Always remember that the tangent to a circle at a point is perpendicular to the radius at that point.