MARS: Angle theorems

The ‘Angle theorems’ resource is a concept development lesson from the ‘Mathematics Assessment Resource Service’. The resource develops conceptual understanding relating to the interior and exterior angles of polygons.

The resource begins with a formative assessment task ‘Four Pentagons’. The task is to calculate the angles marked in the diagram above.


A notable feature of the resource is the support it offers to teachers, with detailed solutions and helpful suggestions for how to help students having difficulty with particular parts of the task.

One method of calculating angle a is to consider angles at a point:

Using the fact that the angles at a point sum to 360º gives

a = 360 – (2 x 108) = 144º

Turning now to angle b we see that the shape between the pentagons is a parallelogram. Since consecutive angles of a parallelogram are supplementary

b = 180 – 144 = 36º

Calculating angle c makes use of angle b, the internal angles of a pentagon, and the fact that angles at a point sum to 360º. The diagram below shows the situation:

c = 360 – (108 + 108 + 36) = 108º

The resource emphasises the need for discussion to promote understanding. Having worked through ideas in the initial formative task, there is a collaborative task to complete: ‘The Pentagon Problem’. This is shown below:

In addition to the diagram, the resource contains outlines of four methods that students have used to calculate the angle measure of x. Examples of student work following these methods are given.

The task for each of the four pieces of work is to:

·         Explain whether the student’s reasoning is correct and complete.

·         Correct the solution if necessary.

·         Use the method to calculate the measure of angle x.

·         Write down all reasoning in detail.

There is a ‘Geometrical Definitions and Properties’ sheet to help with this. In some instances, parts of proofs are missing, in others incorrect assumptions have been made. The work in this resource is ideal in preparing students for the ‘geometrical reasoning’ part of examinations.

The Mathematics Assessment Resource Service (MARS) is a collaboration between the University of California at Berkeley and the Shell Centre team at the University of Nottingham, with support from the Bill and Melinda Gates Foundation. The team is known around the world for its innovative work in maths education. Previous projects that members of the team have worked on include the DfE Standards Units and the Bowland Maths resources.


Age11-14, 14-16
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