11X1 T07 02 triangle theorems (2010)

Triangle Theorems
A
The angle sum of any triangle is 180

B

C

Triangle Theorems
A

B A  B  C  180 sum ABC  180  

C

Triangle Theorems
A

B A  B  C  180 sum ABC  180  

C
Proof:

D
Triangle Theorems
A
E
B A  B  C  180 sum ABC  180  

C
Proof:
Construct DE||BC passing through A

D
Triangle Theorems
A
E
B A  B  C  180 sum ABC  180  

C
Proof:
DAB  ABC alternate ' s , DE || BC 

D
Triangle Theorems
A
E
B A  B  C  180 sum ABC  180  

C
Proof:
EAC  ACB alternate ' s , DE || BC 

D
Triangle Theorems
A
E
B A  B  C  180 sum ABC  180  

C
Proof:
DAB  BAC  CAE  180 straight DAE  180  

D
Triangle Theorems
A
E
B A  B  C  180 sum ABC  180  

C
Proof:
DAB  BAC  CAE  180 straight DAE  180  

 ABC  BAC  ACB  180

A
The exterior angle of any triangle is equal
to the sum of the two opposite interior
B angles

C
D

A
B angles
ACD  A  B exterior , CAB 
C
D

A
B angles
C
D
Proof:

A
B E angles
C
D
Proof:
Construct CE||BA

A
B E angles
C
D
Proof:
Construct CE||BA
ABC  ECD corresponding ' s , CE || BA

A
B E angles
C
D
Proof:
Construct CE||BA
BAC  ACE alternate ' s , CE || BA

A
B E angles
C
D
Proof:
Construct CE||BA
ACD  ACE  ECD common 

A
B E angles
C
D
Proof:
Construct CE||BA
ACD  ACE  ECD common 
 ACD  ABC  BAC

A
Polygon Theorems
B

C
D

A
Polygon Theorems
B The angle sum of any quadrilateral is 360

C
D

A
Polygon Theorems

A  B  C  D  360 sum ABCD  360  

C
D

A
Polygon Theorems

A  B  C  D  360 sum ABCD  360  

Proof:
C
D

A
Polygon Theorems

A  B  C  D  360 sum ABCD  360  

Proof: sum ABC  180
C
D

A
Polygon Theorems

A  B  C  D  360 sum ABCD  360 
Proof: sum ABC  180 (+)
C sum ADC  180
D

A
Polygon Theorems

A  B  C  D  360 sum ABCD  360 
C sum ADC  180
D
sum ABCD  360

A
Polygon Theorems

A  B  C  D  360 sum ABCD  360 
C sum ADC  180
D

B

C
A
D
E

A
Polygon Theorems

A  B  C  D  360 sum ABCD  360 
C sum ADC  180
D

The angle sum of any pentagon is 540
B

C
A
D
E

A
Polygon Theorems

A  B  C  D  360 sum ABCD  360 
C sum ADC  180
D

B
A  B  C  D  E  540 sum ABCDE  540  

C
A
D
E

A
Polygon Theorems

A  B  C  D  360 sum ABCD  360 
C sum ADC  180
D

B
A  B  C  D  E  540 sum ABCDE  540  

C Proof:
A
D
E

A
Polygon Theorems

A  B  C  D  360 sum ABCD  360 
C sum ADC  180
D

B
A  B  C  D  E  540 sum ABCDE  540  

C Proof:
A sum ABE  180

D
E

A
Polygon Theorems

A  B  C  D  360 sum ABCD  360 
C sum ADC  180
D

B
A  B  C  D  E  540 sum ABCDE  540  

C Proof:
A sum ABE  180
sum BED  180
D
E

A
Polygon Theorems

A  B  C  D  360 sum ABCD  360 
C sum ADC  180
D

B
A  B  C  D  E  540 sum ABCDE  540 
C Proof:
A sum ABE  180 (+)
sum BED  180
D sum BDC  180
E

A
Polygon Theorems

A  B  C  D  360 sum ABCD  360 
C sum ADC  180
D

B
A  B  C  D  E  540 sum ABCDE  540 
C Proof:
A sum ABE  180 (+)
sum BED  180
D sum BDC  180
E
sum ABCDE  540

The angle sum of any polygon is 180n-2  ,


where n is the number of sides




a
e b

c
d




a
e b
The exterior angle sum of
any polygon is 360
c
d




a
e b
c
d

a  b  c  d  e  360 exterior sum  360  




a
e b
c
d

a  b  c  d  e  360 exterior sum  360  

Exercise 8B; 1dg, 2c, 3dh, 5ace, 6ab (iii), 7b, 8bfh, 9ad, 10dh,
11ad, 12c, 16, 18, 20

11X1 T07 02 triangle theorems (2010)

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11X1 T07 02 triangle theorems (2010)