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Let's solve the given problem step by step.

### Information from the diagram:
- \( P\hat{B}C = 73^\circ \)
- \( A\hat{O}C = 92^\circ \)
- \( TA \) and \( TB \) are tangents to the circle from \( T \) to points \( A \) and \( B \).
- \( O \) is the center of the circle.

### Part (a) \( A\hat{B}T \)

Since \( TA \) and \( TB \) are tangents from the same point \( T \), the angle between the tangents at point \( B \) is equal to the angle subtended by the opposite arc \( ACB \) at point \( P \). 

By the **tangent-chord theorem**, the angle \( A\hat{B}T \) is equal to \( P\hat{B}C \) (the angle subtended by chord \( BC \) at \( P \)).

\[
A\hat{B}T = P\hat{B}C = 73^\circ
\]

### Part (b) \( B\hat{C}O \)

The total angle \( A\hat{O}C = 92^\circ \) is given, and since \( O\hat{C}B \) and \( B\hat{C}O \) are isosceles, the sum of these two equal angles must add up with \( A\hat{O}C \) to make \( 180^\circ \) at point \( C \) (since it is the center of the circle).

Therefore, the two angles \( B\hat{C}O \) and \( O\hat{C}B \) must be:
\[
B\hat{C}O = O\hat{C}B = \frac{180^\circ - 92^\circ}{2} = 44^\circ
\]

### Part (c) \( A\hat{D}C \)

Using the property of a cyclic quadrilateral, the opposite angles add up to \( 180^\circ \).

Therefore, in quadrilateral \( AODC \), angle \( A\hat{D}C \) is the exterior angle to angle \( A\hat{O}C \). Hence:
\[
A\hat{D}C = 180^\circ - A\hat{O}C = 180^\circ - 92^\circ = 88^\circ
\]

### Final Answers:
- (a) \( A\hat{B}T = 73^\circ \)
- (b) \( B\hat{C}O = 44^\circ \)
- (c) \( A\hat{D}C = 88^\circ \)

Would you like more details or have any questions? 

Here are 5 related questions:
1. What is the tangent-chord theorem, and how does it apply in this context?
2. Why are angles subtended by the same arc equal in a circle?
3. What is a cyclic quadrilateral, and what are its properties?
4. How do tangents from a common external point relate to each other in geometry?
5. How would the angles change if the circle's center was not known?

**Tip**: In cyclic quadrilaterals, always remember that opposite angles sum to \( 180^\circ \). This property is often key to solving angle problems in circles.

In the diagram, TA and TB are tangents to the circle, with center O, at A and B respectively. Given that ∠PBC = 73° and ∠AOC = 92°, calculate:
(a) ∠ABT
(b) ∠BCO
(c) ∠ADC

This problem explores circle theorems with tangents, chords, and cyclic quadrilaterals. Learn how to calculate angles using key theorems like the Tangent-Chord Theorem, Cyclic Quadrilateral Theorem, and more. Suitable for Grades 9-10.

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