Download - 11X1 T07 04 converse theorems
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
CABC 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
CABC 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
CABC 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
CABC 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 Qα α
Converse Theorems(1) The circle whose diameter is the hypotenuse of a right angled
triangle passes through the third vertex.
A B
CABC 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 Qα α 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
CABC 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 Qα α 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