# Question of the Week 5: Circle Theorems

The following question is taken from my website Diagnostic Questions. Here you will find 1000s of high quality maths multiple choice diagnostic / hinge questions, ideal for assessment for learning, which have been created and shared by maths teachers all over the world.

Circle Theorems are a notoriously troublesome topic for many students. Personally I love the challenge of trying to spot which theorems have been used, and trying to fill in the missing angles one step at a time. But that could well be due to the fact that I am a massive maths geek who loves any kind of puzzle, and I have come to realise throughout my teaching career that this is not a character trait shared by the majority of adolescents.

I wrote this question specifically to use with my lovely Year 11 class in the build up to their recent GCSE. We had covered all the theorems a few times, but many of them were still have difficulty spotting which ones to use. Perhaps unsurprisingly, the Alternate Segment Theorem was proving the least popular of the theorems amongst the class, and as I looked through their work in their books I was noticing that they were misapplying the rule left, right and centre. Specifically, they had real trouble identifying which two angles were equal.

So, I wrote this question, and I was pleased to see that it split the class.

Here is what I learnt from their responses and the class discussion that followed:

students seemed to be muddling up the tangent/radius theorem that states the tangent meets a radius at 90 degrees, and thus there were 20 degrees left over.

the most common reason for this were a misapplication of the alternate segment theorem, or thinking that the lines were parallel and hence alternate angles were equal.

this revealed some muddled reasoning about opposite angles in a cyclic quadrilateral