This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

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Here are the calculations for each angle with reasons:
8.1.1 Step 1: Identify the angle as . Step 2: Apply the Tangent-Chord Theorem. The angle between a tangent (TM) and a chord (MN) through the point of contact (M) is equal to the angle in the alternate segment subtended by the chord. Reason: Angle between tangent and chord is equal to the angle in the alternate segment.
**8.1.
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8.1.1 P Step 1: Identify the angle P as MPN. Step 2: Apply the Tangent-Chord Theorem.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.