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Home > Topdrawer > Geometric reasoning > Misunderstandings > Revealing the invisible

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Revealing the invisible

Revealing the invisible

Adding construction lines to a diagram is often necessary when solving problems in geometry. Knowing when and where to add auxiliary lines is difficult for many students.

Drawing an extra (or auxiliary) line in a diagram can reveal 'invisible' relationships. This strategy is particularly helpful when deriving the properties of quadrilaterals, the angle sum of a triangle, the angle sums of polygons and area formulae.

For example, there are several different auxiliary lines that could be added to a trapezium diagram when deriving the formula for its area. Some are illustrated below.

G_CM_M4_fig1

Although some are helpful, others are not. It is valuable for students to explore each situation and determine its merits.

Experimenting with different solution strategies provides students with models for adding auxiliary lines. This helps them see relationships in diagrams beyond the obvious so that they can introduce auxiliary lines without prompting.

A good example of adding extra construction lines can be found when proving the angle sum of a triangle.

You can download the Two Proofs of the Angle Sum of a Triangle PDF. There are further ideas in the Helping to visualise part of the resource.

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Name Class Section
Document An angle in the bend Folder 17
Document Approaching a geometric problem Folder 17
Document Adding auxilliary lines Folder 17
Document More construction lines Folder 17
Document Year 7: Demonstrate that the angle sum of a triangle is 180° and use this to find the angle sum of a quadrilateral. Infobox 3
Document Year 8: Find perimeters and areas of parallelograms, trapeziums, rhombuses and kites Infobox 3