Circle Theorems (Including Proofs)

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Parts
A Reminder about parts of the Circle
Major Segment
diameter
chord
Minor Segment
Major Sector
Minor Sector
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Termgy
Introductory Terminology
Arc AB subtends angle x at the centre.
Arc AB subtends angle y at the circumference.
Chord AB also subtends angle x at the centre.
Chord AB also subtends angle y at the
circumference.
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Th1
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The angle subtended at the centre of a circle (by
an arc or chord) is twice the angle subtended at
the circumference by the same arc or chord.
(angle at centre)
Watch for this one later.
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42o (Angle at the centre).
70o(Angle at the centre)
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(180 2 x 42) 96o (Isos triangle/angle sum
triangle).
48o (Angle at the centre)
124o (Angle at the centre)
(180 124)/2 280 (Isos triangle/angle sum
triangle).
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Th2
This is just a special case of Theorem 1 and is
referred to as a theorem for convenience.
90o angle in a semi-circle
90o angle in a semi-circle
20o angle sum triangle
90o angle in a semi-circle
60o angle sum triangle
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Th3
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Angle x 30o
Angle x angle y 38o
Angle y 40o
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Th4
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If OT is a radius and AB is a tangent, find the
unknown angles, giving reasons for your answers.
180 (90 36) 54o Tan/rad and angle sum of
triangle.
90o angle in a semi-circle
60o angle sum triangle
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Th5
45o (Alt Seg)
60o (Alt Seg)
75o angle sum triangle
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Th6
The opposite angles of a cyclic quadrilateral are
supplementary. (They sum to 180o)
Angles p q 180o
Angles x w 180o
Angles y z 180o
Angles r s 180o
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180 85 95o (cyclic quad)
180 135 45o (straight line)
180 70 110o (cyclic quad)
180 110 70o (cyclic quad)
180 45 135o (cyclic quad)
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Th7
From any point outside a circle only two tangents
can be drawn and they are equal in length.
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PQ and PT are tangents to a circle with centre O.
Find the unknown angles giving reasons.
yo
Q
xo
O
90o (tan/rad)
98o
90o (tan/rad)
49o (angle at centre)
360o 278 82o (quadrilateral)
wo
zo
T
P
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PQ and PT are tangents to a circle with centre O.
Find the unknown angles giving reasons.
zo
Q
O
yo
90o (tan/rad)
xo
180 140 40o (angles sum tri)
50o (isos triangle)
50o (alt seg)
80o
wo
50o
T
P
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Th8
OS 5 cm (pythag triple 3,4,5)
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Angle SOT 22o (symmetry/congruenncy)
Angle x 180 112 68o (angle sum triangle)
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Mixed Q 1
65o (Alt seg)
130o (angle at centre)
25o (tan rad)
25o (isos triangle)
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Mixed Q 2
22o (cyclic quad)
68o (tan rad)
44o (isos triangle)
68o (alt seg)
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Worksheet 3
The angle subtended by an arc or chord at the
centre of a circle is twice the angle subtended
at the circumference by the same arc or chord.
A
O
B
C
Theorem 1 and 2
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Worksheet 4
Angles subtended by an arc or chord in the same
segment are equal.
A
Theorem 3
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Worksheet 5
The angle between a tangent and a radius drawn to
the point of contact is a right angle.
O
B
T
A
Theorem 4
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Worksheet 6
The angle between a tangent and a chord through
the point of contact is equal to the angle
subtended by the chord in the alternate segment.
D
B
O
C
T
A
Theorem 5
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The opposite angles of a cyclic quadrilateral are
supplementary (Sum to 180o).
B
A
C
D
Theorem 6
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Worksheet 8
The two tangents drawn from a point outside a
circle are of equal length.
Theorem 7
P
Theorem 8
A line, drawn perpendicular to a chord and
passing through the centre of a circle, bisects
the chord.
C