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.

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.

Outcomes

9.G.C.2

Equal chords of a circle subtend equal angles at the centre and (motivate) its converse. The perpendicular from the center of a circle to a chord bisects the chord and conversely, the line drawn through the center of a circle to bisect a chord is perpendicular to the chord.There is one and only one circle passing through three given non-collinear points. Equal chords of a circle (or of congruent circles) are equidistant from the center(s) and conversely.

9.G.C.3

The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. Angles in the same segment of a circle are equal. if a line segment joining two points subtends equal angle at two other points lying on the same side of the line containing the segment, the four points lie on a circle. The sum of the either pair of the opposite angles of a cyclic quadrilateral is 180 degree and its converse.

Proofs and Theorems involving Circles