Circle Theorems

Students learn how to recognise and prove various circle theorems including: angle at the centre is double the angle at the circumference, angles in the same segment are equal, opposite angles in cyclic quadrilaterals add to 180°, a tangent runs perpendicular to the radius and opposite angles in alternate segments are equal.  This unit takes place in Term 6 of Year 10 and follows on from Angle Geometry.

Circle Theorems Lessons

Prerequisite Knowledge

  • Understand and use alternate and corresponding angles on parallel lines;
  • Derive and use the sum of angles in a triangle (e.g. to deduce use the angle sum in any polygon, and to derive properties of regular polygons)
  • Measure line segments and angles in geometric figures, including interpreting maps and scale drawings and use of bearings

Success Criteria

  • Identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment
  • Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results

Key Concepts

  • Students need a solid understanding of the properties for angles in parallel lines, vertically opposite, angles in a polygon and on a straight line.
  • Understanding the various parts of a circle is critical to fully defining the various circle theorems.
  • Students need to spend time breaking down the problem by considering the various angle properties that may be relevant.
  • Taking time to prove the various theorems illustrates how interconnected all the properties are.
  • Encourage students to annotate and draw on the diagrams.

Common Misconceptions

  • Students often struggle with precisely defining the various angle the appropriate angle properties.
  • Incomplete angle properties are a common source for losing marks in examinations.
  • Angle and line notation often confuses students to an extent they may calculate an angle that was not asked for.
  • Students need to relate their written work with the relevant angle rather than writing detached paragraphs.


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