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# Circumference Angles

*We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of this resource.*

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Age 11 to 16

Challenge Level

*Circumference Angles printable sheet - proof sorter
Circumference Angles printable sheet - corollaries*

Try moving the points $A$, $B$ and $C$ around (but keep them in the order $A, B, C$ going clockwise!).

What do you notice about the two angles marked in the diagram?

Can you use the diagram below to prove your findings?

**You might like to use this Proof Sorter to help you construct a proof.**

Having now proved this result, you can use it to show that some other things are also true.

Propositions which follow on from a theorem are sometimes called **corollaries**.

**Corollary 1**

Can you show that the angle in a semi-circle is a right-angle?

(I.e. if $AB$ is a diameter then angle $ACB=90^{\circ}$.)

**Corollary 2**

Can you show that the two angles marked in the diagram below are equal?

**Corollary 3**

Can you show that the two angles marked in the diagram below add up to $180^{\circ}$?

Can you use your results to show that the two angles marked in the diagram below add up to $180^{\circ}$?