Transcript
Page 1: 11X1 T07 04 converse theorems

Converse Theorems

Page 2: 11X1 T07 04 converse theorems

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

triangle passes through the third vertex.

Page 3: 11X1 T07 04 converse theorems

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

triangle passes through the third vertex.

A B

C

Page 4: 11X1 T07 04 converse theorems

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

Page 5: 11X1 T07 04 converse theorems

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 =∠

Page 6: 11X1 T07 04 converse theorems

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.

Page 7: 11X1 T07 04 converse theorems

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

Page 8: 11X1 T07 04 converse theorems

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

Page 9: 11X1 T07 04 converse theorems

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 =∠′

Page 10: 11X1 T07 04 converse theorems

(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.

Page 11: 11X1 T07 04 converse theorems

(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

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(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

Page 13: 11X1 T07 04 converse theorems

(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 =∠′

Page 14: 11X1 T07 04 converse theorems

(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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