The document discusses finding the angle between two lines given by their gradients (slopes). It provides the formula for calculating the tangent of the acute angle between the lines as (m1 - m2) / (1 + m1m2), where m1 and m2 are the gradients of the two lines. It then works through three examples of applying the formula to find the acute angle between pairs of lines.

1. Angle between 2 Lines
Preliminary Extension Mathematics
Date: Tuesday 10th May 2011

2. Angle between 2 lines
y
line l1 has gradient m1
l2 l1 line l2 has gradient m2
∴ m1 = tan α and m2 = tan β
θ
α β
x
0

3. Angle between 2 lines
y
line l1 has gradient m1
l2 l1 line l2 has gradient m2
∴ m1 = tan α and m2 = tan β
θ
α β
x
and α +θ =β (Why?)
0

4. Angle between 2 lines
y
line l1 has gradient m1
l2 l1 line l2 has gradient m2
∴ m1 = tan α and m2 = tan β
θ and α +θ =β
α β
x (Exterior angle of V)
0

5. Angle between 2 lines
y
So
l2 l1 θ = β −α
θ
α β
x
0

6. Angle between 2 lines
y
So
l2 l1 θ = β −α
∴ tan θ = tan(β − α )
θ
α β
x
0

7. Angle between 2 lines
y
So
l2 l1 θ = β −α
∴ tan θ = tan(β − α )
θ tan β − tan α
=
α β 1 + tan β tan α
x
0
You will learn this formula later

8. Angle between 2 lines
y
So
l2 l1 θ = β −α
∴ tan θ = tan(β − α )
θ tan β − tan α
=
α β 1 + tan β tan α
x
0
m1 − m2
=
1 + m1m2
Why?

9. Angle between 2 lines
y
So
l2 l1 θ = β −α
∴ tan θ = tan(β − α )
θ tan β − tan α
=
α β 1 + tan β tan α
x
0
m1 − m2
=
1 + m1m2
When tan θ is positive, θ is acute.
When tan θ is negative, θ is obtuse.

10. Angle between 2 lines
y
Thus for two lines of gradient
l2 l1 m1 and m2
the acute angle between them is given by
θ m1 − m2
tan θ =
α β
x 1 + m1m2
0
Note that m1m2 ≠ −1 what does this mean?

11. Angle between 2 lines
y
Thus for two lines of gradient
l2 l1 m1 and m2
the acute angle between them is given by
θ m1 − m2
tan θ =
α β
x 1 + m1m2
0
Note that m1m2 ≠ −1
the formula does not work for perpendicular lines

12. Example 1
Find the acute angle between y = 2x + 1 and y = −3x − 2
(to nearest degree)

13. Example 1
Find the acute angle between y = 2x + 1 and y = −3x − 2
(to nearest degree)
∴ m1 = 2 and m2 = −3

14. Example 1
Find the acute angle between y = 2x + 1 and y = −3x − 2
(to nearest degree)
∴ m1 = 2 and m2 = −3
m1 − m2
tan θ =
1 + m1m2

15. Example 1
Find the acute angle between y = 2x + 1 and y = −3x − 2
(to nearest degree)
∴ m1 = 2 and m2 = −3
m1 − m2
tan θ =
1 + m1m2
2+3
∴ tan θ =
1− 6

16. Example 1
Find the acute angle between y = 2x + 1 and y = −3x − 2
(to nearest degree)
∴ m1 = 2 and m2 = −3
m1 − m2
tan θ =
1 + m1m2
2+3
∴ tan θ =
1− 6
∴ tan θ = −1

17. Example 1
Find the acute angle between y = 2x + 1 and y = −3x − 2
(to nearest degree)
∴ m1 = 2 and m2 = −3
m1 − m2
tan θ =
1 + m1m2
2+3
∴ tan θ =
1− 6
∴ tan θ = −1
∴ tan θ = 1

18. Example 1
Find the acute angle between y = 2x + 1 and y = −3x − 2
(to nearest degree)
∴ m1 = 2 and m2 = −3
m1 − m2
tan θ =
1 + m1m2
2+3
∴ tan θ =
1− 6
∴ tan θ = −1
∴ tan θ = 1 → θ = 45°

19. Example 2
Find the acute angle between 3x − 2y + 7 = 0 and
(to nearest degree) 2y + 4x − 3 = 0

20. Example 2
Find the acute angle between 3x − 2y + 7 = 0 and
(to nearest degree) 2y + 4x − 3 = 0
∴2y = 3x + 7

21. Example 2
Find the acute angle between 3x − 2y + 7 = 0 and
(to nearest degree) 2y + 4x − 3 = 0
∴2y = 3x + 7
3 7
∴y = x +
2 2

22. Example 2
Find the acute angle between 3x − 2y + 7 = 0 and
(to nearest degree) 2y + 4x − 3 = 0
∴2y = 3x + 7
3 7
∴y = x +
2 2
3
∴ m1 =
2

23. Example 2
Find the acute angle between 3x − 2y + 7 = 0 and
(to nearest degree) 2y + 4x − 3 = 0
∴2y = 3x + 7
similarly ∴2y = −4x + 3
3 7
∴y = x + 3
2 2 ∴ y = −2x +
2
3
∴ m1 = ∴ m2 = −2
2

24. Example 2
Find the acute angle between 3x − 2y + 7 = 0 and
(to nearest degree) 2y + 4x − 3 = 0
applying the formula
m1 − m2
tan θ = 3
1 + m1m2 m1 = m2 = −2
2

25. Example 2
Find the acute angle between 3x − 2y + 7 = 0 and
(to nearest degree) 2y + 4x − 3 = 0
applying the formula
m1 − m2
tan θ = 3
1 + m1m2 m1 = m2 = −2
2
3
+2
∴ tan θ = 2
1− 3

26. Example 2
Find the acute angle between 3x − 2y + 7 = 0 and
(to nearest degree) 2y + 4x − 3 = 0
applying the formula
m1 − m2
tan θ = 3
1 + m1m2 m1 = m2 = −2
2
3
+2
∴ tan θ = 2
1− 3
−7
∴ tan θ =
4

27. Example 2
Find the acute angle between 3x − 2y + 7 = 0 and
(to nearest degree) 2y + 4x − 3 = 0
applying the formula
m1 − m2
tan θ = 3
1 + m1m2 m1 = m2 = −2
2
3
+2
∴ tan θ = 2
1− 3
−7 7
∴ tan θ = ∴ tan θ = → θ = 60°
4 4

28. Example 3 - by thinking and
drawing....
Find the acute angle between y= x+3 and y = −3x + 5
(to nearest degree)

29. Example 3 - by thinking and
drawing....
Find the acute angle between y= x+3 and y = −3x + 5
(to nearest degree)
y = −3x + 5 y
y= x+3
θ
α β
x
0

30. Example 3 - by thinking and
drawing....
Find the acute angle between y= x+3 and y = −3x + 5
(to nearest degree)
m1 = 1 → α = 45°
y = −3x + 5 y
y= x+3
θ
α β
x
0

31. Example 3 - by thinking and
drawing....
Find the acute angle between y= x+3 and y = −3x + 5
(to nearest degree)
m1 = 1 → α = 45°
y = −3x + 5 y
y= x+3 m2 = −3 → β = 108°
θ
α β
x
0

32. Example 3 - by thinking and
drawing....
Find the acute angle between y= x+3 and y = −3x + 5
(to nearest degree)
m1 = 1 → α = 45°
y = −3x + 5 y
y= x+3 m2 = −3 → β = 108°
But α +θ =β
θ
α β
x
0

33. Example 3 - by thinking and
drawing....
Find the acute angle between y= x+3 and y = −3x + 5
(to nearest degree)
m1 = 1 → α = 45°
y = −3x + 5 y
y= x+3 m2 = −3 → β = 108°
But α +θ =β
θ ∴θ = 63°
α β
x
0