Year 10 - Maths
The opposite angles of a cyclic quadrilateral sum to 180°
Circle theorems
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This pupil-friendly study guide includes a unit summary, clear notes, common mistakes, and quick questions with answers.
Unit Summary
In this unit, pupils will use their developed mathematical reasoning skills to derive and use various circle theorems.
Lesson Summary
You will learn to derive and use the theorem: the opposite angles of a cyclic quadrilateral sum to 180°.
Key Notes
- A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments
- Theorems can be thought of as puzzles to solve, you are showing how to find a result
- In order to use this theorem, you may need to draw a diagram or add more information to an existing one
Vocabulary To Learn
- Cyclic quadrilateral: A cyclic quadrilateral is a quadrilateral where all four of its vertices lie on a single circle.
Common Mistakes To Avoid
- Pupils may conflate cyclic quadrilaterals with cases of "the angle at the centre of a circle is twice the angle at the circumference" theorem, when the angle at the centre is a reflex angle.
3 Quick Questions (With Answers)
1. Solve this in steps: explain the method from this lesson and why each step matters.
A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments. Theorems can be thought of as puzzles to solve, you are showing how to find a result.
2. Use this key word in a maths sentence and explain it clearly. 'Cyclic quadrilateral'
A cyclic quadrilateral is a quadrilateral where all four of its vertices lie on a single circle. Use this word when showing your method, then check your final answer is reasonable.
3. Spot and fix this common maths mistake.
Mistake: Pupils may conflate cyclic quadrilaterals with cases of "the angle at the centre of a circle is twice the angle at the circumference" theorem, when the angle at the centre is a reflex angle. Correction: While the two diagrams may look similar at a glance, pay extra attention to whether all four points of the quadrilateral are on the circumference or if one of the points is in the centre.
More Lessons In This Unit
- Checking and securing understanding of the parts of a circle Study Guide
- Identifying which circle theorem to use Study Guide
- Problem solving with circle theorems Study Guide
- The alternate segment theorem Study Guide
- The angle at the centre of the circle is twice the angle at any point on the circumference Study Guide
- The angle in a semicircle is a right angle Study Guide
- The angles in the same segment are equal Study Guide
- The perpendicular from the centre of a circle to a chord bisects the chord Study Guide
- The tangent at any point on a circle is perpendicular to the radius at that point Study Guide
- The tangents from an external point are equal in length Study Guide
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