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Tangent-chord angle equals the angle in the alternate segment

This type of question, asked 4 times: 19 Jul 2023 S2, 12 Sep 2024 S3, 13 Sep 2024 S3, 25 Sep 2024 S3. All 4 questions below, oldest first.

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  1. Question 1

    19 Jul 2023 S2

    52
    Two circles touch each other externally at P. AB is a direct common tangent to the two circles. If A and B are points of contact and ∠PAB=65∘\angle PAB = 65^\circ, then ∠ABP\angle ABP is _________.

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  2. Question 2

    12 Sep 2024 S3

    51
    A pair of straight lines from an external point F intersects a circle at A and B (FA < FB), and touches the circle at C. O is the centre of the circle. Given that ∠ACF=50∘\angle ACF = 50^\circ and ∠AFC=30∘\angle AFC = 30^\circ, find ∠AOB\angle AOB.

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  3. Question 3

    13 Sep 2024 S3

    56
    Two circles, centred at P and Q intersect at two points C and D. AB is tangent to the two circles at A and B. If ∠ADB=68∘\angle ADB = 68^\circ, then ∠ACB=\angle ACB = __________.

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  4. Question 4

    25 Sep 2024 S3

    72
    From a point T, a tangent TP at point P, is drawn to a circle with centre O. A secant TQR (point Q is near to point T) is drawn from the point T. ΔPQR\Delta PQR is inscribed into the circle by joining the points P, Q and R. Draw lines OQ and OR. If ∠PTQ\angle PTQ is 27∘27^\circ and ∠TPQ=55∘\angle TPQ = 55^\circ, what is the degree measure of ∠ROQ\angle ROQ?

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