Give a reason for each statement: (a) Q1 = arT

Here are the reasons for each statement based on the provided diagram and information.

2.1 Give a reason for each statement.

• Statement (a): $Q_1 = \hat{T}$

  Step 1: Identify the relevant geometric theorem.
  $Q_1$ is the angle between the tangent PQ and the chord QS of the circle QST. $\hat{T}$ (or $\angle QTS$) is the angle subtended by the chord QS in the alternate segment of the circle QST.

  Reason (a): Tangent-Chord Theorem

  $$\boxed{\text{}Q_1 = \hat{T} \quad \text{Reason: Tangent-Chord Theorem}\text{}}$$

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