Angles, Polygons and Geometrical Proof: Age 14-16

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

Why does this fold create an angle of sixty degrees?

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

Is it possible to find the angles in this rather special isosceles triangle?

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

Two ladders are propped up against facing walls. At what height do the ladders cross?

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

Can you work out the fraction of the original triangle that is covered by the inner triangle?

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

Can you work out the side length of a square that just touches the hypotenuse of a right angled triangle?

A napkin is folded so that a corner coincides with the midpoint of an opposite edge. Investigate the three triangles formed.

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

Using a ruler, pencil and compasses only, it is possible to construct a square inside any triangle so that all four vertices touch the sides of the triangle.

What is the same and what is different about these circle questions? What connections can you make?

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

Can you make sense of the three methods to work out what fraction of the total area is shaded?