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Point on a side with an angle relation: find an angle by the exterior angle theorem

This type of question, asked 2 times: 29 Jan 2022 Tier II, 3 Feb 2022 Tier II. All 3 questions below, oldest first.

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  1. Question 1

    29 Jan 2022 Tier II

    16
    In △PQR\triangle PQR, SS is a point on the side QRQR such that ∠QPS=12∠PSR\angle QPS = \frac{1}{2}\angle PSR, ∠QPR=78∘\angle QPR = 78^\circ and ∠PRS=44∘\angle PRS = 44^\circ. What is the measure of ∠PSQ\angle PSQ?

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  2. Question 2

    3 Feb 2022 Tier II

    96
    In △PQR\triangle PQR, PQ = PR and S is a point on QR such that ∠PSQ=96∘+∠QPS\angle PSQ = 96^\circ + \angle QPS and ∠QPR=132∘\angle QPR = 132^\circ. What is the measure of ∠PSR\angle PSR?

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  3. Question 3

    3 Feb 2022 Tier II

    81
    In △ABC\triangle ABC, D is a point on BC such that ∠ADB=2∠DAC\angle ADB = 2\angle DAC, ∠BAC=70∘\angle BAC = 70^\circ and ∠B=56∘\angle B = 56^\circ. What is the measure of ∠ADC\angle ADC?

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