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## Circle Theorems 1

### Angles in the same segment are equal

The dotted line divides the circle into two segments.  This line is called a chord.

Angles in the same chord as their start and finish and that are in the same segment are equal.

Examples

Find the values of x and y

Angle $x$ has the same start and finish as $83^o\mbox{, so }x=83^o$

Angle $y$ has the same start and finish as $33^o\mbox{, so }x=33^o$

Find the values x,y and zAngle $x$ has the same start and finish as $63^o\mbox{, so }x=83^o$ Angle $y=180^o-81^o-63^o=36^o\because 180^o\mbox{ in a triangle}$Angle $z$ has the same start and finish as $y\mbox{, so }z=36^o$

### Angle at the centre is twice the angle at the circumference

Like the theorem above both of these angles need to be in the same segment

#### Proof

Examples

Find the value of x

Since the angle at the centre is twice the angle at the circumference $x=\frac{120^o}{2}=60^o$

Find the value of x

Since the angle at the centre is twice the angle at the circumference $x=58^o\times 2=116^o$

### Angle in a semi-circle

The angle in a semi-circle is always $90^o$

Opposite angles in a cyclic quadrilateral

The opposite angles in a cyclic quadrilateral are supplementary (add up to $180^o$)

Example

Find the value of $x$

Since PQRS is a cyclic quadrilateral $\angle RQP = 99^o\because$ angles in a cyclic quad are supplementary.

$\implies x= 81^o\because$ angles on a line add up to $180^o$