Angles - points, lines and parallel lines

Weekly Problem 16 - 2007
Can you figure out how far the robot has travelled by the time it is first facing due East?

P is a point inside a square ABCD such that PA= 1, PB = 2 and PC = 3. How big is angle APB ?

Six circles around a central circle make a flower. Watch the flower as you change the radii in this circle packing. Prove that with the given ratios of the radii the petals touch and fit perfectly.

A spiropath is a sequence of connected line segments end to end taking different directions. The same spiropath is iterated. When does it cycle and when does it go on indefinitely?

Weekly Problem 9 - 2012
What is the angle QPT in this diagram?

Have a good look at these images. Can you describe what is happening? There are plenty more images like this on NRICH's Exploring Squares CD.

Weekly Problem 39 - 2010
If you know three lengths and an angle in this diagram, can you find another angle by calculation?

Consider a watch face which has identical hands and identical marks for the hours. It is opposite to a mirror. When is the time as read direct and in the mirror exactly the same between 6 and 7?

Use simple trigonometry to calculate the distance along the flight path from London to Sydney.

Weekly Problem 44 - 2010
Extend two of the sides of a nonagon to form an angle. How large is this acute angle?