Geometrical Reasoning

Creating convincing arguments or "proofs" to show that statements are always true is a key mathematical skill. The problems in this feature offer you the chance to explore geometrical properties, make conjectures and create proofs to show that these are always true.

Many of the problems in this feature include proof sorting activities which challenge you to rearrange statements in order to recreate clear, rigorous proofs.  

Plus magazine has a selection of interesting articles exploring proofs in which pictures play an important role.

 

Circumference Angles 

Age 11 to 16
Challenge Level
Can you prove the angle properties described by some of the circle theorems?

Cyclic Quadrilaterals Proof 

Age 11 to 16
Challenge Level
Can you prove that the opposite angles of cyclic quadrilaterals add to $180^\circ$?

Pythagoras Proofs 

Age 11 to 16
Challenge Level
Can you make sense of these three proofs of Pythagoras' Theorem?

Matter of Scale 

Age 14 to 16
Challenge Level
Can you prove Pythagoras' Theorem using enlargements and scale factors?

Overlap 

Age 14 to 16
Challenge Level
A red square and a blue square overlap. Is the area of the overlap always the same?

Pentakite 

Age 14 to 18
Challenge Level
Given a regular pentagon, can you find the distance between two non-adjacent vertices?

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?

Kite in a Square 

Age 14 to 18
Challenge Level
Can you make sense of the three methods to work out what fraction of the total area is shaded?

The Converse of Pythagoras 

Age 14 to 18
Challenge Level
Can you prove that triangles are right-angled when $a^2+b^2=c^2$?


We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of these resources.