1. Angle Between Two Lines

2. Angle Between Two Lines
y
x

3. Angle Between Two Lines
y
1
x
l1

4. Angle Between Two Lines
y
l1 has slope m1  tan 1
1
x
l1

5. Angle Between Two Lines
l
y 2
l1 has slope m1  tan 1
2 1
x
l1

6. Angle Between Two Lines
l
y 2
l1 has slope m1  tan 1
l2 has slope m2  tan  2
2 1
x
l1

7. Angle Between Two Lines
l
y 2
l1 has slope m1  tan 1
 l2 has slope m2  tan  2
2 1
x
l1

8. Angle Between Two Lines
l
y 2
l1 has slope m1  tan 1
 l2 has slope m2  tan  2
2 1 1     2 (exterior  )
x
l1

9. Angle Between Two Lines
l
y 2
l1 has slope m1  tan 1
 l2 has slope m2  tan  2
2 1 1     2 (exterior  )
x
  1   2
l1

10. Angle Between Two Lines
l
y 2
l1 has slope m1  tan 1
 l2 has slope m2  tan  2
2 1 1     2 (exterior  )
x
  1   2
tan   tan 1   2 
l1

11. Angle Between Two Lines
l
y 2
l1 has slope m1  tan 1
 l2 has slope m2  tan  2
2 1 1     2 (exterior  )
x
  1   2
tan   tan 1   2 
l1
tan 1  tan  2

1  tan 1 tan  2

12. Angle Between Two Lines
l
y 2
l1 has slope m1  tan 1
 l2 has slope m2  tan  2
2 1 1     2 (exterior  )
x
  1   2
tan   tan 1   2 
l1
tan 1  tan  2

1  tan 1 tan  2
m1  m2

1  m1m2

13. Angle Between Two Lines
l y 2
l1 has slope m1  tan 1
 l2 has slope m2  tan  2
2 1 1     2 (exterior  )
x
  1   2
tan   tan 1   2 
l1
tan 1  tan  2

1  tan 1 tan  2
m1  m2

1  m1m2
The acute angle between two lines with slopes m1 and m2 can be
found using;

14. Angle Between Two Lines
l y 2
l1 has slope m1  tan 1
 l2 has slope m2  tan  2
2 1 1     2 (exterior  )
x
  1   2
tan   tan 1   2 
l1
tan 1  tan  2

1  tan 1 tan  2
m1  m2

1  m1m2
The acute angle between two lines with slopes m1 and m2 can be
found using; m1  m2
tan  
1  m1m2

15. 2000 Extension 1 HSC Q1b)
e.g. Find the acute angle between the lines
1
y  2 x  1 and y  x  1
3

16. 2000 Extension 1 HSC Q1b)
e.g. Find the acute angle between the lines
1
y  2 x  1 and y  x  1
3
1
2
tan   3
1
1  2  
 3

17. 2000 Extension 1 HSC Q1b)
e.g. Find the acute angle between the lines
1
y  2 x  1 and y  x  1
3
1
2
tan   3
1
1  2  
 3
6 1

3 2

18. 2000 Extension 1 HSC Q1b)
e.g. Find the acute angle between the lines
1
y  2 x  1 and y  x  1
3
1
2
tan   3
1
1  2  
 3
6 1

3 2
1

19. 2000 Extension 1 HSC Q1b)
e.g. Find the acute angle between the lines
1
y  2 x  1 and y  x  1
3
1
2
tan   3
1
1  2  
 3
6 1

3 2
1
  45

20. 2005 Extension 1 HSC Q1f)
(ii) The acute angle between the lines y = 3x + 5 and y = mx + 4 is 45
Find the possible values of m.

21. 2005 Extension 1 HSC Q1f)
(ii) The acute angle between the lines y = 3x + 5 and y = mx + 4 is 45
Find the possible values of m.
3 m
tan 45 

1  3m

22. 2005 Extension 1 HSC Q1f)
(ii) The acute angle between the lines y = 3x + 5 and y = mx + 4 is 45
Find the possible values of m.
3 m
tan 45 

1  3m
3 m
1
1  3m

23. 2005 Extension 1 HSC Q1f)
(ii) The acute angle between the lines y = 3x + 5 and y = mx + 4 is 45
Find the possible values of m.
3 m
tan 45 

1  3m
3 m
1
1  3m
1  3m  3  m

24. 2005 Extension 1 HSC Q1f)
(ii) The acute angle between the lines y = 3x + 5 and y = mx + 4 is 45
Find the possible values of m.
3 m
tan 45 

1  3m
3 m
1
1  3m
1  3m  3  m
1  3m  3  m or  1  3m   3  m

25. 2005 Extension 1 HSC Q1f)
(ii) The acute angle between the lines y = 3x + 5 and y = mx + 4 is 45
Find the possible values of m.
3 m
tan 45 

1  3m
3 m
1
1  3m
1  3m  3  m
1  3m  3  m or  1  3m   3  m
4m  2

26. 2005 Extension 1 HSC Q1f)
(ii) The acute angle between the lines y = 3x + 5 and y = mx + 4 is 45
Find the possible values of m.
3 m
tan 45 

1  3m
3 m
1
1  3m
1  3m  3  m
1  3m  3  m or  1  3m   3  m
4m  2
1
m
2

27. 2005 Extension 1 HSC Q1f)
(ii) The acute angle between the lines y = 3x + 5 and y = mx + 4 is 45
Find the possible values of m.
3 m
tan 45 

1  3m
3 m
1
1  3m
1  3m  3  m
1  3m  3  m or  1  3m   3  m
4m  2  1  3m  3  m
1
m
2

28. 2005 Extension 1 HSC Q1f)
(ii) The acute angle between the lines y = 3x + 5 and y = mx + 4 is 45
Find the possible values of m.
3 m
tan 45 

1  3m
3 m
1
1  3m
1  3m  3  m
1  3m  3  m or  1  3m   3  m
4m  2  1  3m  3  m
1
m 2m  4
2

29. 2005 Extension 1 HSC Q1f)
(ii) The acute angle between the lines y = 3x + 5 and y = mx + 4 is 45
Find the possible values of m.
3 m
tan 45 

1  3m
3 m
1
1  3m
1  3m  3  m
1  3m  3  m or  1  3m   3  m
4m  2  1  3m  3  m
1
m 2m  4
2
m  2

30. 2005 Extension 1 HSC Q1f)
(ii) The acute angle between the lines y = 3x + 5 and y = mx + 4 is 45
Find the possible values of m.
3 m
tan 45 

1  3m
3 m
1
1  3m
1  3m  3  m
1  3m  3  m or  1  3m   3  m
4m  2  1  3m  3  m
1
m 2m  4
2
1 m  2
 m  or m  2
2

31. Exercise 14E; 1ac, 2bdf, 4, 7, 8, 10, 11a, 15*