T2W2L1: Circle Geometry

Chapter 7: Circle Geometry

• in this chapter there will be a lot of geometric proofs.

geometric proofs §

• focus is on euclidean geometry
• assumed to know
  • angle sum theorems for triangles & quadrilaterals.
  • properties of intersecting lines (congruence of vertically opposite angles).
  • relationships involving angles in parallel lines with a transversal (corresponding, alternate interior etc etc.)
  • isosceles triangle theorem (angles opposite congruent sides in a triangle are themselves congruent).
  • triangle congruence and similarity ‘tests’ (SSS, SAS, etc)

circles §

• a circle is the set of all points at a given distance r from a given point $0$ called the centre.
• two points on a circle defines two arcs:
  • if the two points are not diametrically opposite each other, then there exists a smaller and bigger arc called the
    • **minor arc
    • major arc**
  • if it is diametrically opposite each other, then there exists two semi circles.
  • points drawn from the centre to the edges of an arc is called a sector.
    • minor sector, major sector
  • if you join the circle with a straight line (called a chord), it is called a segment.
• subtending: the angle constructed by the arc.
  • subtending does not have to occur only in a sector, a minor arc AB can subtend the angle ACB at the ‘circumference’ of the circle

circle theorems §

1. an angle at the centre is twice the angle of the circumference §

• prove that $\angle AOC=2\times \angle ABC$
• proof:
  • $OA=OB$ (radii)
  • $\therefore △AOB$ is isosceles.
  • $\angle OAB=\angle OBA=x°$ ($△AOB$ is isosceles)
  • $\angle AOD=2x°$ (exterior angle is sum of remote interior angles.)
  • theorem: in a triangle, an exterior angle is the sum of the 2 remote interior angles.
    • c = 180 [sum of a and b]
    • c = 180 - [exterior angle]

2. angles in a semicircle are right angles §

• refer to edwards note :D

3. angles in the same segment are equal §

• uh
• a cyclic quadrilateral is a quadrilateral whose vertices all line on the same circle

4. opposite angles of a cyclical quadrilateral are supplementary §