State and prove the alternate segment theorem.

1. 1 · Write the theorem A line touches a circle, and a chord is drawn from the point of contact. Then each angle between the tangent and the chord equals the angle in the alternate segment.
2. 2 · Write what is given and what is to be proved Given: tangent PQ touches the circle with centre O at A; AB is a chord; C is on the major arc. To prove: ∠BAQ = ∠ACB.
3. 3 · Draw the figure Circle, tangent PQ at A, chord AB, point C on the major arc, and diameter AD. Join BD, BC and AC.
4. 4 · Use the two right angles
   • ∠DAQ = 90° (tangent ⊥ radius), so ∠BAQ = 90° − ∠DAB
   • ∠ABD = 90° (angle in a semicircle), so ∠ADB = 90° − ∠DAB
5. 5 · Compare So ∠BAQ = ∠ADB.
6. 6 · Use the same segment ∠ADB = ∠ACB (angles in the same segment).
7. 7 · Check and box the answer ∠BAQ = ∠ACB. Hence proved. In the same way, ∠BAP equals the angle in the other segment.

Scoring tip: Draw the diameter AD clearly, because both right angles come from it. Write the reason in brackets each time.