UK Secondary (7-11)
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Proofs and Theorems involving Circles

We've already looked at some different rules applying to circle geometry. In this chapter, we are going to review these rules and look at how we can use them for different proofs.

When you are proving a statement is true, you can use any geometrical rules you know about circles, such as the radii in a circle are equal, angle properties and properties of quadrilaterals. You just need to select work out which facts support your proof.  


Circle geometry theorems & rules

  • Angle at the Centre Theorem: An inscribed angle $\left(a\right)$(a) is half of the central angle $\left(2a\right)$(2a)
  • Angles Subtended by Same Arc Theorem: When there are two fixed endpoints, the angle $\left(a\right)$(a) is always the same, no matter where it is on the circumference.
  • Angle in a Semicircle: The angle inscribed in a semicircle is always a right angle ($90^\circ$90°).
  • Cyclic Quadrilaterals: The opposite angles in a cyclic quadrilateral add up to $180^\circ$180°.


Now let's use these to solve some proofs.


Worked Examples

Question 1

Consider the figure to the right, in which $O$O is the centre of the circle, and $\angle AOB=\angle DOC$AOB=DOC.

Prove that $\triangle AOB$AOB and $\triangle DOC$DOC are congruent.

  1. In $\triangle ABO$ABO and $\triangle CDO$CDO we have:.

Question 2

In the diagram, $O$O is the centre of the circle. Show that $x$x and $y$y are supplementary angles.

Question 3

Prove $AB$AB = $DE$DE.




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