11X1 T13 04 converse theorems (2010)

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  • Converse Theorems

  • Converse Theorems (1) The circle whose diameter is the hypotenuse of a right angled

    triangle passes through the third vertex.

  • Converse Theorems (1) The circle whose diameter is the hypotenuse of a right angled

    triangle passes through the third vertex.

    A B

    C

  • Converse Theorems (1) The circle whose diameter is the hypotenuse of a right angled

    triangle passes through the third vertex.

    A B

    C ABC are concyclic with AB diameter

  • Converse Theorems (1) The circle whose diameter is the hypotenuse of a right angled

    triangle passes through the third vertex.

    A B

    C ABC are concyclic with AB diameter

    90 semicircle ain

  • Converse Theorems (1) The circle whose diameter is the hypotenuse of a right angled

    triangle passes through the third vertex.

    A B

    C ABC are concyclic with AB diameter

    90 semicircle ain

    (2) If an interval AB subtends the same angle at two points P and Q on

    the same side of AB, then A,B,P,Q are concyclic.

  • Converse Theorems (1) The circle whose diameter is the hypotenuse of a right angled

    triangle passes through the third vertex.

    A B

    C ABC are concyclic with AB diameter

    90 semicircle ain

    (2) If an interval AB subtends the same angle at two points P and Q on

    the same side of AB, then A,B,P,Q are concyclic.

    A B

    P Q

  • Converse Theorems (1) The circle whose diameter is the hypotenuse of a right angled

    triangle passes through the third vertex.

    A B

    C ABC are concyclic with AB diameter

    90 semicircle ain

    (2) If an interval AB subtends the same angle at two points P and Q on

    the same side of AB, then A,B,P,Q are concyclic.

    A B

    P Q ABQP is a cyclic quadrilateral

  • Converse Theorems (1) The circle whose diameter is the hypotenuse of a right angled

    triangle passes through the third vertex.

    A B

    C ABC are concyclic with AB diameter

    90 semicircle ain

    (2) If an interval AB subtends the same angle at two points P and Q on

    the same side of AB, then A,B,P,Q are concyclic.

    A B

    P Q ABQP is a cyclic quadrilateral

    aresegment samein s

  • (3) If a pair of opposite angles in a quadrilateral are supplementary (or

    if an exterior angle equals the opposite interior angle) then the

    quadrilateral is cyclic.

  • (3) If a pair of opposite angles in a quadrilateral are supplementary (or

    if an exterior angle equals the opposite interior angle) then the

    quadrilateral is cyclic.

    A

    D

    B

    C

    60

    120

  • (3) If a pair of opposite angles in a quadrilateral are supplementary (or

    if an exterior angle equals the opposite interior angle) then the

    quadrilateral is cyclic.

    A

    D

    B

    C

    60

    120 ABCD are concyclic

  • (3) If a pair of opposite angles in a quadrilateral are supplementary (or

    if an exterior angle equals the opposite interior angle) then the

    quadrilateral is cyclic.

    A

    D

    B

    C

    60

    120 ABCD are concyclic

    180 are ralquadrilate cyclicin sopposite

  • (3) If a pair of opposite angles in a quadrilateral are supplementary (or

    if an exterior angle equals the opposite interior angle) then the

    quadrilateral is cyclic.

    A

    D

    B

    C

    60

    120 ABCD are concyclic

    180 are ralquadrilate cyclicin sopposite

    Exercise 9D; 1, 2, 3, 6b, 7b, 10a, 11