# Angles in a Circle

In these lessons, we will give a review and summary of the properties of angles that can be formed in a circle and their theorems.

- Inscribed angles subtended by the same arc are equal.
- Central angles subtended by arcs of the same length are equal.
- The central angle of a circle is twice any inscribed angle subtended by the same arc.
- Angle inscribed in semicircle is 90˚.
- An angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.
- The opposite angles of a cyclic quadrilatreral are supplementary
- The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.
- A radius or diameter that is perpendicular to a chord divides the chord into two equal parts and vice versa.
- A tangent to a circle is perpendicular to the radius drawn to the point of tangency.
- When two segments are drawn tangent to a circle from the same point outside the circle, the segments are equal in length.

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More Geometry Lessons
## Inscribed angles subtended by the same arc are equal

The following diagram shows inscribed angles subtended by the same arc are equal.

∠* x* = ∠* y* because they are subtended by the same arc *AEC*.

## Central angles subtended by arcs of the same length are equal

The following diagram shows central angles subtended by arcs of the same length are equal.

∠* x* = ∠* y* if arc *AB* = arc *CD *

## The Central Angle is twice the Inscribed Angle

The following diagrams show the central angle of a circle is twice any inscribed angle subtended by the same arc.

## Angle inscribed in semicircle is 90°

The following diagram shows the angle inscribed in semicircle is 90 degrees.

*POQ* is the diameter. ∠*PAQ = ∠PBQ = ∠PCQ *= 90˚.

## Alternate Segment Theorem

The diagram shows an angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

The alternate segment theorem tells us that ∠*CEA *= ∠*CDE*

## Angles in a Cyclic Quadrilateral

In a cyclic quadrilateral, the opposite angles are supplementary i.e. they add up to 180°

* a* + *c* = 180 ˚**, ***b* + *d* = 180 ˚

## Exterior angle of a cyclic quadrilateral is equal to the interior opposite angle

The following diagram shows the exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

The exterior angle ∠* ADF* is equal to the corresponding interior angle ∠* ABC. *

The exterior angle ∠* DCE* is equal to the corresponding interior angle ∠* DAB. *

## Radius perpendicular to a chord bisects the chord

A radius or diameter that is perpendicular to a chord divides the chord into two equal parts and vice versa.

In the above circle, if the radius *OB* is perpendicular to the chord *PQ* then *PA* = *AQ*.

## Tangent to a Circle Theorem

A tangent to a circle is perpendicular to the radius drawn to the point of tangency.

## Two-Tangent Theorem

When two segments are drawn tangent to a circle from the same point outside the circle, the segments are equal in length.

In the following diagram:

If *AB* and *AC* are two tangents to a circle centred at *O*, then:

- the tangents to the circle from the external point
*A* are equal
*OA* bisects the angle *BAC *between the two tangents
*OA* bisects the angle *BOC *between the two radii to the points of contact
- triangle
*AOB *and triangle *AOC* are congruent right triangles

## Videos

This video gives a review of the following circle theorems: arrow theorem, bow theorem, cyclic quadrilateral, semi-circle, radius-tangent theorem, alternate segment theorem, chord center theorem, dual tangent theorem.

This video gives a review of the following circle theorems: same segment, subtended by arc, angle in semicircle, tangents equal length, radius tangent, alternate segment, bisect chord, cyclic quadrilateral. It also includes the proofs of the theorem.