In the above figure, PQR is a tangent at Q and RST is a straight line. QTU = 50°, UTW = 33° and TRQ = 25°. Giving reasons in each case, calculate (i) Angle TSQ (ii) Angle TSW (iii) Angle TUQ (iv) Angle PQW

$\angle QTW$ is the sum of $\angle QTU$ and $\angle UTW$.

Reason: Angles on a straight line (if U is on TW) or Angle addition.

By the Alternate Segment Theorem, the angle between the tangent PQR and the chord QS ($\angle RQS$) is equal to the angle subtended by the chord QS in the alternate segment ($\angle QTW$).

Reason: Alternate Segment Theorem.

Consider $\triangle RQS$. The sum of angles in a triangle is $180^\circ$.

Since S is on the line RST, $\angle TSQ = \angle RSQ$.

Reason: Sum of angles in a triangle.
The angle TSQ is $\boxed{72^\circ}$.

a) (ii) Angle TSW

Angles subtended by the same arc are equal. $\angle QSU$ and $\angle QTU$ are both subtended by arc QU.

Reason: Angles subtended by the same arc.

Since S is on the line RST, $\angle TSQ = \angle TSU + \angle QSU$.

Reason: Angle subtraction.

Angles subtended by the same arc are equal. $\angle TQU$ and $\angle TSU$ are both subtended by arc TU.

Reason: Angles subtended by the same arc.

Angles subtended by the same arc are equal. $\angle UQW$ and $\angle UTW$ are both subtended by arc UW.

Reason: Angles subtended by the same arc.

$\angle TQW$ is the sum of $\angle TQU$ and $\angle UQW$.

Reason: Angle addition.

Angles subtended by the same arc are equal. $\angle TSW$ and $\angle TQW$ are both subtended by arc TW.

Reason: Angles subtended by the same arc.
The angle TSW is $\boxed{55^\circ}$.

a) (iii) Angle TUQ

By the Alternate Segment Theorem, the angle between the tangent PQR and the chord QT ($\angle PQT$) is equal to the angle subtended by the chord QT in the alternate segment ($\angle QWT$).
First, find $\angle PQT$.
$\angle PQT = \angle QST$. We found $\angle QST = \angle TSQ = 72^\circ$ in part (i).
So, $\angle PQT = 72^\circ$.
Therefore, $\angle QWT = 72^\circ$.
Reason: Alternate Segment Theorem.

Angles subtended by the same arc are equal. $\angle TUQ$ and $\angle QWT$ are both subtended by arc TQ.

Reason: Angles subtended by the same arc.
The angle TUQ is $\boxed{72^\circ}$.

a) (iv) Angle PQW

By the Alternate Segment Theorem, the angle between the tangent PQR and the chord QW ($\angle PQW$) is equal to the angle subtended by the chord QW in the alternate segment ($\angle QTW$).
We found $\angle QTW = 83^\circ$ in part (i).

Reason: Alternate Segment Theorem.
The angle PQW is $\boxed{83^\circ}$.