Problem Analysis

From the diagram and description:

1. $OA$ is the diameter of the circle, so $\angle OBA = 90^\circ$ (Thales' theorem).
2. $\angle OAC = 38^\circ$ is given.
3. $TC$ is a tangent to the circle at $C$. By the tangent-secant theorem, the angle between the tangent and a chord equals the angle subtended by the chord at the opposite segment.

We need to determine the following angles with explanations:

(a) $\angle DCT$

• Since $TC$ is a tangent to the circle at $C$, $\angle DCT$ equals the angle subtended by $CD$ at the opposite segment of the circle.
• $\angle DCT = \angle CBA$ (Alternate Segment Theorem).

To find $\angle CBA$:

• In $\triangle OAC$, $\angle OAC = 38^\circ$, and $\angle OBA = 90^\circ$.
• Since the sum of angles in a triangle is $180^\circ$:
  $\angle ACB = 180^\circ - (90^\circ + 38^\circ) = 52^\circ.$ Thus: $\angle DCT = \angle CBA = 52^\circ.$

(b) $\angle DEA$

• $\angle DEA$ subtends the diameter $OA$, so it forms a right angle (semicircle property): $\angle DEA = 90^\circ.$

(c) $\angle ACB$

• Already calculated above: $\angle ACB = 52^\circ.$

(d) $\angle BDC$

• The angle $BDC$ subtends the same arc $BC$ as $\angle BAC$.
• Since angles subtending the same arc are equal: $\angle BDC = \angle BAC = 38^\circ.$

(e) $\angle BOA$

• $BOA$ is the central angle subtended by arc $BA$.
• The central angle is twice the angle subtended at the circumference by the same arc.
• $\angle BAC = 38^\circ$, so: $\angle BOA = 2 \times 38^\circ = 76^\circ.$

Final Answers:

1. $\angle DCT = 52^\circ$
2. $\angle DEA = 90^\circ$
3. $\angle ACB = 52^\circ$
4. $\angle BDC = 38^\circ$
5. $\angle BOA = 76^\circ$

Do you need further clarification or detailed steps?
Here are some related questions to expand your understanding:

1. How does the tangent-segment theorem help solve problems involving tangents?
2. What is Thales' theorem, and how does it apply to diameters of circles?
3. How do you prove that the angle subtended by a diameter is always $90^\circ$?
4. How can you determine central angles from angles at the circumference?
5. What are alternate segment angles, and how are they used in tangent problems?

Tip: Remember, angles subtended by the same chord in a circle are always equal!