Circle Geometry

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1 Theorem - Equal arcs on circles of equal radii subtend equal angles at the centre. Theorem - Equal angles at the centre of a circle on equal arcs. Info Go
 

Objective: On completion of the lesson the student will be able to prove that 'Equal arcs on circles of equal radii, subtend equal angles at the centre', and that 'Equal angles at the centre of a circle stand on equal arcs.' They should then be able to use these pro

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2 Theorem - The perpendicular from the centre of a circle to a chord bisects the chord. Theorem - The line from the centre of a circle to the mid-point of the chord is perpendicular to the chord. Info Go
 

Objective: On completion of the lesson the student will be able to prove that 'The perpendicular from the centre of a circle to a chord bisects the chord.' and its converse theorem 'The line from the centre of a circle to the mid-point of the chord is perpendicular'

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3 Theorem - Equal chords in equal circles are equidistant from the centres. Theorem - Chords in a circle which are equidistant from the centre are equal. Info Go
 

Objective: On completion of the lesson the student will be able to prove that equal chords in equal circles are equidistant from the centre.

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4 Theorem - The angle at the centre of a circle is double the angle at the circumference standing on the same arc. Info Go
 

Objective: On completion of the lesson the student will be able to prove that the angle at the centre of a circle is double the angle at the circumference standing on the same arc.

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5 Theorem - Angles in the same segment of a circle are equal. Info Go
 

Objective: On completion of the lesson the student will be able to prove that the angles in the same segment are equal.

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6 Theorem - The angle of a semi-circle is a right angle. Info Go
 

Objective: On completion of the lesson the student will be able to prove that 'The angle of a semi-circle is a right-angle.'

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7 Theorem - The opposite angles of a cyclic quadrilateral are supplementary. Info Go
 

Objective: On completion of the lesson the student will be able to prove that the opposite angles of a cyclic quadrilateral are supplementary.

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8 Theorem - The exterior angle at a vertex of a cyclic quadrilateral equals the interior opposite angle. Info Go
 

Objective: On completion of the lesson the student will be able to prove that the exterior angle at a vertex of a cyclic quadrilateral equals the interior opposite.

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9 Theorem - The tangent to a circle is perpendicular to the radius drawn to it at the point of contact. Info Go
 

Objective: On completion of the lesson the student will be able to prove that the tangent and the radius of a circle are perpendicular at the point of contact.

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10 Theorem - Tangents to a circle from an external point are equal. Info Go
 

Objective: On completion of the lesson the student will be able to prove that tangents to a circle from an external point are equal.

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11 Theorem - The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. Info Go
 

Objective: On completion of the lesson the student will be able to prove that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

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12 Theorem - When circles touch, the line of the centres passes through the point of contact. Info Go
 

Objective: On completion of the lesson the student will be able to prove that ' When two circles touch, the line of the centres passes through the point of contact', and use this result to complete questions that require it.

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