Properties of Angles in Circles

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    PROPERTIES OFANGLES IN CIRCLES

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    Angles At Centre and Circumference

    An angle at the centre ofa circle is an angle withits vertex at the centreand two radii as its

    sides.

    An angle at thecircumference is an

    angle with its vertex atthe circumference of thecircle and two chords asits sides.

    Q

    P

    C

    A

    AQC

    APC

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    Example:

    Name the angles that are subtended by arc PQ at

    the centre and at the circumference respectively.a) b)

    QP

    y

    x

    O

    g

    f

    e

    S

    R

    Q

    P

    O

    Angles at the centre

    = ________

    Angle at the circumference

    = ________

    Angles at the centre

    = ________

    Angle at the circumference

    = ________

    y

    x

    e

    f

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    Angles Subtended By Same Arc

    Angles subtended atthe circumference bythe same arc are

    equal.z

    y

    x

    Q

    P

    x = y = z because

    these angles aresubtended by the same

    arc PQ (a.k.aangles in

    the same segment )

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    Example:

    Name the angles that are of the same size.

    a) b)

    O

    zu

    y

    x

    w

    zx

    y

    O

    x = y y = z

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    Angles Subtended By Arcs

    Of Same Length

    The angles subtended at the circumference by arcs of

    the same length are equal.

    R

    P

    Q

    yx

    B

    C

    A

    When length of minor arc AC = length

    of minor arc PR, then x = y.

    When x = y, then length of minor arc AC =length of minor arc PR.

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    The angles subtended at the centre by arcs of the samelength are equal.

    O

    y

    x

    Q

    P

    B

    A

    When length of minor arc ab =

    length of minor arc PQ, then x = y.

    When x = y, then length of minor

    arc AB = length of minor arc PQ.

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    Example:

    Name the angles that are of the same size in each

    of the following.a) b)

    xy

    O

    u

    z w

    v

    u

    O

    x = y v = w

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    Relationship Between Angles

    Angle subtended by an arc at its centre, istwice the angle at the circumference.

    Angle at the centre

    = 2 x Angle at the circumference

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    Example:

    O

    2x

    x

    O

    2x

    x

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    Example:

    2x

    x

    x

    OO

    2x

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    Example:

    Find the value of x.

    a)

    x

    35O

    x = 2 x 35

    x = 70

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    b)

    xO

    120

    x = 2 x 120

    x = 240

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    c)

    x60

    O

    x =60

    2

    x = 30

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    d)

    x

    280

    O

    x =280

    2

    x = 140

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    Angles In A Semicircle

    Any angle at the circumference of a semicircle is aright angle (90).

    Example:

    Q

    P

    BAO

    Since AOB is the diameter;then APB =AQB =90.

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    Example:

    Find the value of x.

    a)

    x38

    x = 180 - 90 - 38x = 58

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    b)

    xx =

    180 - 90

    2x = 45

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    c)

    x = 90 - 42

    x = 48

    42

    x

    O

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    Solving ProblemsExample:

    1)

    C

    B

    A

    110

    X

    O

    In the figure, O is thecentre of the circle.

    Find the value of x.Reflex angle AOC= 2 x 110

    = 220x = 360 - 220x = 140

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    2)

    O

    60

    x

    D

    B

    CA

    In the figure, O is thecentre of the circle.Find the value of x.

    ABC = 90CBD = 90 - 60

    = 30

    x = 30(Angles in thesame segment)

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    3)

    T

    D

    C

    B

    A30

    x

    O

    In the figure, O is the centreof the circle. ATC and BOD arestraight lines. Find the valueof x.

    BOC = 2 x 50= 60

    x = 180 - 60= 120

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    4)T

    S

    R Q P

    x

    110

    50

    O

    In the figure, SOQ is thediameter of the circle

    with centre O. Find thevalue of x.

    RSQ = 50

    QRS = 90

    RQS = 180 - 90 - 50= 40

    x = 180 -110 - 40= 30