11x1 t13 03 angle theorems 2 (2011)

Angle Theorems

Angle Theorems (5b) Opposite angles of a cyclic quadrilateral are supplementary.

Angle Theorems (5b) Opposite angles of a cyclic quadrilateral are supplementary.

C

A

B

O

D

Angle Theorems (5b) Opposite angles of a cyclic quadrilateral are supplementary.

180 ralquadrilate cyclicin s'opposite 180 ADCABC

C

A

B

O

D

Angle Theorems (5b) Opposite angles of a cyclic quadrilateral are supplementary.

180 ralquadrilate cyclicin s'opposite 180 ADCABC

180 :Prove

C

A

B

O

D

Angle Theorems (5b) Opposite angles of a cyclic quadrilateral are supplementary.

180 ralquadrilate cyclicin s'opposite 180 ADCABC

180 :Prove Proof: Join AO and CO

C

A

B

O

D

Angle Theorems (5b) Opposite angles of a cyclic quadrilateral are supplementary.

180 ralquadrilate cyclicin s'opposite 180 ADCABC

180 :Prove Proof: Join AO and CO

arc samenceon circumfere

at twicecentreat 2αAOC

C

A

B

O

D

Angle Theorems (5b) Opposite angles of a cyclic quadrilateral are supplementary.

180 ralquadrilate cyclicin s'opposite 180 ADCABC

180 :Prove Proof: Join AO and CO

arc samenceon circumfere

at twicecentreat 2αAOC

C

A

B

O

D

arc samenceon circumfere

at twicecentreat 2reflexAOC

Angle Theorems (5b) Opposite angles of a cyclic quadrilateral are supplementary.

180 ralquadrilate cyclicin s'opposite 180 ADCABC

180 :Prove Proof: Join AO and CO

arc samenceon circumfere

at twicecentreat 2αAOC

revolution 36022 reflexAOCAOC

C

A

B

O

D

arc samenceon circumfere

at twicecentreat 2reflexAOC

Angle Theorems (5b) Opposite angles of a cyclic quadrilateral are supplementary.

180 ralquadrilate cyclicin s'opposite 180 ADCABC

180 :Prove Proof: Join AO and CO

arc samenceon circumfere

at twicecentreat 2αAOC

revolution 36022 reflexAOCAOC

180

C

A

B

O

D

arc samenceon circumfere

at twicecentreat 2reflexAOC

(5c) The exterior angle of a cyclic quadrilateral is equal to the

opposite interior angle.

(5c) The exterior angle of a cyclic quadrilateral is equal to the

opposite interior angle.

C

A

B

O

D

X

interior opposite

alquadilater cyclicexterior BADDCX

(5c) The exterior angle of a cyclic quadrilateral is equal to the

opposite interior angle.

C

A

B

O

D

X

(6) Angles subtended at the circumference by the same or equal arcs

(or chords) are equal.

(6) Angles subtended at the circumference by the same or equal arcs

(or chords) are equal.

A

B

O

C

D

(6) Angles subtended at the circumference by the same or equal arcs

(or chords) are equal.

aresegment samein s' ACDABD

A

B

O

C

D

(6) Angles subtended at the circumference by the same or equal arcs

(or chords) are equal.

aresegment samein s' ACDABD

ACDABD :Prove

A

B

O

C

D

(6) Angles subtended at the circumference by the same or equal arcs

(or chords) are equal.

aresegment samein s' ACDABD

ACDABD :Prove

Proof: Join AO and DO

A

B

O

C

D

(6) Angles subtended at the circumference by the same or equal arcs

(or chords) are equal.

aresegment samein s' ACDABD

ACDABD :Prove

Proof: Join AO and DO

arc sameon ncecircumfere

at twicecentreat 2 ABDAOD

A

B

O

C

D

(6) Angles subtended at the circumference by the same or equal arcs

(or chords) are equal.

aresegment samein s' ACDABD

ACDABD :Prove

Proof: Join AO and DO

arc sameon ncecircumfere

at twicecentreat 2 ABDAOD

A

B

O

C

D

arc sameon ncecircumfere

at twicecentreat 2 ACDAOD

(6) Angles subtended at the circumference by the same or equal arcs

(or chords) are equal.

aresegment samein s' ACDABD

ACDABD :Prove

Proof: Join AO and DO

arc sameon ncecircumfere

at twicecentreat 2 ABDAOD

ACDABD

A

B

O

C

D

arc sameon ncecircumfere

at twicecentreat 2 ACDAOD

(6) Angles subtended at the circumference by the same or equal arcs

(or chords) are equal.

aresegment samein s' ACDABD

ACDABD :Prove

Proof: Join AO and DO

arc sameon ncecircumfere

at twicecentreat 2 ABDAOD

ACDABD

Exercise 9C; 1 ace etc, 2 aceg, 3, 4 bd, 6 bdf, 7 bdf, 9, 13, 14

A

B

O

C

D

arc sameon ncecircumfere

at twicecentreat 2 ACDAOD