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The alternate segment theorem

Lesson details

Learning outcome

I can derive and use the theorem: the alternate segment theorem.

Key learning points

  1. A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments
  2. Theorems can be thought of as puzzles to solve, you are showing how to find a result
  3. In order to use this theorem, you may need to draw a diagram or add more information to an existing one

Keywords

  • Theorem - A theorem is a statement that can be proved using known mathematical facts and reasoning.

  • Subtended - An angle can be subtended by a line segment or curve. The legs of the subtended angle meet the endpoints of the line segment or curve.

  • Segment - Two segments are created when dividing a circle into two parts using a chord.

  • Tangent - A tangent of a circle is a line that intersects the circle exactly once.

Common misconception

Pupils may conflate the alternate segment theorem with alternate angles on parallel lines, thinking that the chord that meets the tangent is a transversal between two equal angles.

Alternate angles are only equal when the transversal is between parallel lines. In many of the examples for the alternate segment theorem, the tangent is not parallel to a chord.

Teacher tip

The lesson demonstrates how this circle theorem could be explored practically by using a circular geoboard and using dynamic geometry software. Pupils may wish to explore these further for themselves.

Licence

This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

Lesson video

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Prior knowledge starter quiz

6 Questions

Q1.
Find the size of angle $$x$$°.

An image in a quiz
Correct answer: 29°
30.5°
61°
90°
Not enough information.

Q2.
Calculate the size of angle $$m$$°.

An image in a quiz
Correct Answer: 75, 75°, 75 degrees

Q3.
Both ∠GHF and ∠GIF are angles at the circumference subtended by the chord GF. The size of the angle labelled $$p$$° is °.

An image in a quiz
Correct Answer: 34, 34°, 34 degrees

Q4.
Both ∠JMK and ∠JLK are angles at the circumference subtended by the arc JK. The size of the angle labelled $$t$$° is °.

An image in a quiz
Correct Answer: 76, 76°, 76 degrees

Q5.
The size of the angle labelled $$a$$° is °.

An image in a quiz
Correct Answer: 28, 28°, 28 degrees

Q6.
Find the size of angle $$b$$° (in degrees).

An image in a quiz
Correct Answer: 64, 64°, 64 degrees

6 Questions

Q1.
Find the size of angle $$a$$° (in degrees).

An image in a quiz
Correct answer: 35°
55°
90°
110°
Not enough information.

Q2.
Find the size of angle $$b$$° (in degrees).

An image in a quiz
Correct Answer: 55, 55°, 55 degrees

Q3.
Find the size of angle $$c$$° (in degrees).

An image in a quiz
Correct Answer: 55, 55°, 55 degrees

Q4.
The size of the angle labelled $$d$$° is °.

An image in a quiz
Correct Answer: 48, 48°, 48 degrees

Q5.
∠ATC = 46° ∠ABT is subtended by the chord AT. Find the size of angle $$e$$° (in degrees).

An image in a quiz
Correct Answer: 46, 46°, 46 degrees

Q6.
∠PEF is subtended by the chord PF. ∠PFE is subtended by the chord PE. Find ∠EPD (in degrees).

An image in a quiz
Correct Answer: 84, 84°, 84 degrees

To help you plan your 10 maths lesson on: The alternate segment theorem, download all teaching resources for free and adapt to suit your pupils' needs...