Angle properties related to circles

General Information
 'Deduce and apply the angle properties related to circles.' is a bullet point under Measurement of Geometry and Measurement in Mathematics and Statistics, Level 6. This is usually assessed in either a simple question format requiring some calculation, or as part of a more complex question with multiple steps.

What you need to know
Many angle properties relate to circles, some of which have been covered in previous levels. The following are some new ones that you will probably find helpful. These rules may not be of accurate wording for your school/area, so refer to textbooks/other information for what to write in exams. Note that these can be shortened in exams in various ways.

Angles from the same arc are equal: You can tell the arc from which the angle comes by the two points of that angle that are not connected. This applies to angles where all three of the points are on a circle's circumference.

Angle in a semicircle is 90 degrees: By 'semicircle', this property means a 180 degree arc. Applies to the same angles as above.

The angle at the center is two time the angle at the circumference: Suppose you had an arc. If the third point was at the center of the circle, that angle would be twice as large as it would be if the third point was on the circumference.

Opposite angles of a cylic quadrilateral add to 180 degrees: A cylic quadrilateral is a quadrilateral (shape with four sides) with all four points on a circle's perimeter. Opposite angles (angles not adjacent to each other in this case) of such a quadrilateral will always add to 180 degrees.

Tangent is at 90 degrees to the radius: A tangent is a line that intersects with a point on the circle's perimeter. Note that (obviously) this only applies to one radius for each tangent, and not all of them.

Triangles with radii are isosceles: Triangles with two radii of the same circle as sides will always be isoceles.

Two useful symbols when it comes to geometric reasoning are '∵' to represent 'because', ‘→' to represent 'and', and '∴' to represent 'Therefore'. This can save you a lot of writing.

For example:

∵∠A=∠B

→∠B=∠C

∴∠A=∠C

Useful links
Due to the difficulty of drawing pictures in a wikia editor and the likely inaccuracy, we have not provided any practice questions for this area. However, the following are some useful links for revision and practice: