11 x1 t13 04 converse theorems (2012)
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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
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