1. Subtending
•In I/GCSE Mathematics, when an angle is created between two lines,
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4936730/292257_134842.png)
•The angle in between the two lines is subtended between the arc C and D
2. Angles at the Centre and Circumference
•The angle subtended at an arc is twice the size of an angle subtended at the circumference
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4936730/470282_731469.png)
•More simply the angle at the centre is double the angle at the circumference
Proof
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4936730/638199_478907.png)
•Let angle OGH = y and angle OGK = x.
• Angle y = angle OHG because GOH is an isosceles.
Lengths OH and OG are both radii.
• Angle x = angle OKG because GOK is an isosceles.
Lengths OK and OG are also radii.
•Angle GOH = 180-2y
Angle GOK = 180- 2x (angles in a triangle add up to 180°)
•Angle JOH = 2y
Angle JOK = 2x (angles on a straight line add up to 180°)
• The angle at the centre KOH (2y+2x) is double the angle at the circumference KGH (x+y).
3. Angles in the Same Segment
•The angle at the circumference subtended by the same arc are equal
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4936730/523455_366069.png)
•Simply the angles in the same segment are equal
•Angles a = a
Proof
•Let the reflex angle MOQ = 2x
Using the circle theorem the angle at the centre is twice the angle at the circumference.
•Angle MNQ = x and angle MPQ = x
Therefore angle MNQ = angle MPQ.
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4936730/564032_731758.png)
4. Cyclic Quadrilaterals
•In I/GCSE Mathematics, a cyclic quadrilateral is a quadrilateral drawn inside a circle where every corner of the quadrilateral must touch the circumference.
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4936730/445847_539467.png)
•The opposite angles in a cyclic quadrilateral add up to 180°
•E.g. a + c = 180°
b + d = 180°
Proof
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4936730/99347_534310.png)
• Let angle CDE = x and angle EFC = y
•!Remember the angle at the centre is double
the angle at the circumference!
•Angle COE = 2y and the reflex angle COE = 2x
•Angles around a point add up to 360°
• 2y + 2x = 360°
• 2y/2 + 2x/2 = 360° /2
• So y and x = 180°