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.

 

Leave a Reply

Your email address will not be published. Required fields are marked *

You may use these HTML tags and attributes:

<a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <s> <strike> <strong>

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Mr Mathematics Blog

Angles in Polygons

There are two key learning points when solving problems with angles in polygons.  The first is to understand why all the exterior angles of a polygon have a sum of 360°.  The second is to understand the interior and exterior angles appear on the same straight line. Students can be told these two facts and […]

Getting Ready for a New School Year

When getting ready for a new school year I have a list of priorities to work through. Knowing my team have all the information and resources they need to teach their students gives me confidence we will start the term in the best possible way.  Mathematics Teaching and Learning Folder All teachers receive a folder […]

Mathematics OFSTED Inspection – The Deep Dive

Earlier this week, my school took part in a trial OFSTED inspection as part of getting ready for the new inspection framework in September 2019. This involved three Lead Inspectors visiting our school over the course of two days. The first day involved a ‘deep dive’ by each of the Lead Inspectors into Mathematics, English […]