More construction lines

The aim of additional construction lines is to introduce a new relationship within the diagram. This can then be used as a step towards other relationships.

To achieve this, one or more of the following constructions may be helpful.

• Draw a diagonal and investigate any triangles which result.
• Draw a parallel line and use the extra corresponding, co-interior or alternate angles which are formed.
• Draw a perpendicular and use the complementary angles which are formed.
  

• Produce a line segment to form an isosceles triangle.
• Produce a line segment to form a midpoint.
• Join ends of line segments and look for similar triangles.
• Draw a radius, diameter or chord in a circle and investigate the angles and triangles formed.

Students find this aspect of problem solving in geometry very challenging. Provide opportunities for them to explore the consequences of adding different auxiliary lines to a diagram. This will encourage a more creative approach to problem solving.