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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 $180^o$