Angles - points, lines and parallel lines

problem
Similarly so

Age

14 to 16

Challenge level

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.
problem
Angle trisection

Age

14 to 16

Challenge level

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
problem
Hand swap

Age

14 to 16

Challenge level

My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the minute hand and hour hand had swopped places. What time did the train leave London and how long did the journey take?
problem
On time

Age

11 to 14

Challenge level

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?
problem
Arrowhead

Age

14 to 16

Challenge level

The points P, Q, R and S are the midpoints of the edges of a non-convex quadrilateral.What do you notice about the quadrilateral PQRS and its area?
problem
Quad in quad

Age

14 to 18

Challenge level

Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?
problem
A problem of time

Age

14 to 16

Challenge level

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?
problem
Flexi quads

Age

16 to 18

Challenge level

A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?
problem
Square world

Age

16 to 18

Challenge level

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

Age

7 to 11

Challenge level

This investigation explores using different shapes as the hands of the clock. What things occur as the the hands move.