# 11X1 T13 04 converse theorems (2011)

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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