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On Time

On a clock the three hands - the second, minute and hour hands - are on the same axis. How often in a 24 hour day will the second hand be parallel to either of the two other hands?

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Estimating Angles

How good are you at estimating angles?

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LOGO Challenge 8 - Rhombi

Explore patterns based on a rhombus. How can you enlarge the pattern - or explode it?

Weekly Problem 35 - 2009

Stage: 3 Short Challenge Level: Challenge Level:1

solution

Since $ABQ$ and $BCQ$ are equilateral, the angles $ABP$ and $CBQ$ are both $60^\circ$. So $$\angle{PBQ} = 360^\circ-90^\circ-60^\circ-60^\circ=150^\circ$$
PBQ is isosceles, so the angles $BPQ$ and $PQB$ are equal. So
$$2 \times \angle{PQB} = 180^\circ- 150^\circ = 30^\circ$$ So $$\angle{PQB} = 15^\circ$$

This problem is taken from the UKMT Mathematical Challenges.

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