Angles in a Circle

In this lesson, 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.

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.

∠ x = ∠ y because arc AB = arc CD

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

Angle inscribed in semicircle is 90˚.

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

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 angle CEA and angle CDE are equal.

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

a + c = 180 ˚, b + d = 180 ˚

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.

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

 

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

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