The document provides guidance on identifying and calculating angles between lines and planes in 3-dimensional space. It outlines four key skills: 1) Identifying the angle between a line and plane, 2) Calculating the angle between a line and plane, 3) Identifying the angle between two planes, and 4) Calculating the angle between two planes. Examples are given to demonstrate how to use trigonometric functions like tangent to determine specific angles within diagrams of 3D objects. Activities are also included for students to practice applying the skills, such as identifying angles within diagrams of cuboids.

2. LINES AND PLANES IN 3-DIMENSION
To answer the question from this topic, the
students must acquire the following skills :
• Able to identify the angle between a line
and a plane ( 1 Mark )
• Able to calculate the angle between a line
and a plane ( 3 Marks ).
• Able to identify the angle between two
planes ( 1 Mark ).
• Able to calculate the angle between two
planes ( 3 Marks )

3. EXAMPLE :
Diagram shows a cuboid with
W
T P1 a horizontal rectangular base.
5cm  Calculate the angle between
U V the planeTWR and the plane
S PSWT.
P
8cm
SOLUTION :
Q R
 R T/W S
8
Tan  RWS = 
5 K2
At the back
W 8
 RWS = Tan -1
5
5cm
S R = 580  N1
8cm

4. ACTIVITY 1 : TO IDENTIFY THE PLANE
H G
E F
D C
A
B
PLANE AT THE TOP : PLANE EFGH

5. H G
E F
D C
A
B
PLANE ON THE LEFT : PLANE ADHE

6. H G
E F
D C
A
B
PLANE IN THE FRONT : PLANE ABFE

7. H G
E F
D C
A
B
PLANE AT THE BACK : PLANE DCGH

8. H G
E F
D C
A
B
PLANE AT THE BOTTOM: PLANE ABCD

9. H G
E F
D C
A
B
PLANE ON THE RIGHT : PLANE BCGF

10. THE LOCATION OF THE POINT
ON TOP OF THE RED DOT
TO THE RIGHT OF
THE RED DOT
AT THE BACK OF
THE RED DOT
IN FRONT OF THE
RED DOT

11. ON THE
TOP OF ….
AT THE
TO THE BACK OF ….
LEFT OF ….
TO THE
RIGHT OF ….
IN FRONT
OF ….
AT THE
BOTTOM OF ….

12. ACTIVITY 2 : TO DETERMINE THE LOCATION OF A POINT
H G
E
F
D C
A B
POINT TO THE LEFT OF F : POINT E
POINT AT THE BOTTOM OF F : POINT B
POINT AT THE BACK OF F : POINT G
POINT TO THE RIGHT OF D : POINT C
POINT ON TOP OF D : POINT H
POINT IN FRONT OF D : POINT A

13. ANGLE BETWEEN
A LINE AND A PLANE
A
LINE
PLANE
C B

14. Activity 3 :To Identify The Angle Between Line And Plane
H G
The line draw from
G and
Normal perpendicular to
the plane ABCD is
E F call normal
D C
The line lies on the
Orthogonal plane ABCD which
projection joint the point A to the
A B line GC is known as
the orthogonal
The angle between the line AG
projection of line AG
and the orthogonal projection, AC
on the plane ABCD.
is the angle between the line AG
and the plane ABCD that is
 GAC.

15. ACTIVITY 3 : To Identify The Angle Between A Line And A Plane
Example 1a

H G
G A C
Normal
E F
D C At the bottom
Name the angle between the line
A B AG and the plane ABCD
Orthogonal
projection
Angle between the line AG and the plane ABCD
= GAC.

16. EXAMPLE 1(b)
H G
E F
D C
A B
Diagram 1(b)
Diagram 1b shows a cuboid ABCDEFGH.
Name the angle between the line HB and the
plane ABCD.

17. ACTIVITY 4 :
To find the angle between a line and a plane
Example 2(a) 12cm
H G
5cm 5cm
E F
D
C
4cm
A B
Diagram 2a
Diagram 2(a) shows a cuboid, ABCDEFG. Find the
angle between the line AH and the plane DCGH.

18. No Steps Solutions
1. Draw the line AH and
shade the plan DCGH H 12cm
G
in diagram 2a.
5cm
E F
D C
4cm
A B
Diagram 2a
Diagram 2a shows a cuboid, ABCDEFG. Find the angle between the line
AH and the plane DCGH.

19. No Steeps Solutions
2 Use the method you
have learned in activity
3, identify the angle
between the line AH  A H D
and the plane DCGH
back
H 12cm
G
5cm
E F
D C
4cm
A B

20. No Steps Solutions
3 Refer to the points you have H
obtained in steep 2 (point A, H,
D), complete the ∆ AHD. Mark
 AHD. Mark the right angle,
 HDA. Transfer out the
A D
∆ AHD.
12cm

H G
A H D
5cm
E F
D C
4cm
A B

21. No Steps Solutions
4 With the information given in the
question, label the length of the
sides of ∆ AHD. At least the
length for 2 sides must be known.
Use Pythegoras Theorem if
necessary.
H 12cm G
5cm
E F
D C
4cm
A B

22. No Steps Solutions
6 Mark, H
- the opposite side, AD asT
- the adjacent side, HD as S
5 cm S
A 4 cm D
T
H 12cm
G
5cm
E F
D C
4cm
A B

23. No Steps Solutions
6 Use the tangent formula to
4
calculate  AHD. Tan  AHD =
5
Remember, use
4
-The sine formula, if O and H were  AHD = tan -1
5
known O - SOH
S  AHD = 38040’
H
- The cosine formula, if A and H
were known
A
C – CAH 12cm
H G
H
-The tangent formula, if O and A E 5cm
D F
were known O C
T – TOA 4cm
A A B

24. example 2 (b)
12 cm
H G
E F
4 cm
D
C
3 cm
A B
Diagram 2b
Diagram 2b shows a cuboid,ABCDEFGH. Calculate
the angle between the line HB and the plane BCGF

25. ANGLE BETWEEN TWO
PLANES

26. ACTIVITY 5 : To Identified The Angle Between Two
Planes
EXAMPLE 3(a)
H G
E
F
D C
A 1. DRAW 3 LINES
B
Diagram 3a
Diagram 3a shows a cuboid,
ABCDEFGH. Name the angle between
the plane AGH and the plane ABCD

27. ACTIVITY 5 : To Identified The Angle Between Two Planes
H G
E F
D C
Bottom
A B
2. Mark the location
Diagram 3a
(direction) of the
plane ABCD at the
Diagram 3a shows a cuboid,
ABCDEFGH. Name the angle bottom of the first
between the plane, AGH and the line to the left.
plane, ABCD

28. ACTIVITY 5 : To Identified The Angle Between Two Planes
H G
A
E F
D C Bottom
A B 3. Refer to the plane,
Diagram 3a AGH, identify the
points which
Diagram 3a shows a cuboid, touch the plane,
ABCDEFGH. Name the angle
between the plane, AGH and the ABCD and write it
plane, ABCD at the middle line.

29. ACTIVITY 5 : To Identified The Angle Between Two Planes
H G
H/G A
E F
D C Bottom
A 3. Refer to the plane,
B
Diagram 3a
AGH, identify the
point which does
Diagram 3a shows a cuboid,
not touch the
ABCDEFGH. Name the angle plane, ABCD and
between the plane, AGH and the write it at the first
plane, ABCD
line to the left.

30. ACTIVITY 5 : To Identified The Angle Between Two Planes
H G
H/G A
E Bottom
F
D C
5. Between the point H
and G, point which is
A B nearer to point A or
Diagram 3a
located on the same
plane as point A will
Diagram 3a shows a cuboid,
ABCDEFGH. Name the angle
be choosen. Point
between the plane, AGH and the which is not choosen
plane, ABCD will be earased.

31. ACTIVITY 5 : To Identified The Angle Between Two Planes
H G
H A
E Ke Bawah
F
D C
5. Between the point H
and G, point which is
A B nearer to point A or
Diagram 3a
located on the same
plane as point A will
Diagram 3a shows a cuboid,
ABCDEFGH. Name the angle
be choosen. Point
between the plane, AGH and the which is not choosen
plane, ABCD will be earased.

32. ACTIVITY 5 : To Identified The Angle Between Two Planes
H G
 H A D
E Bottom
F
D C
6. Identify the point
which is located at
A B the bottom of the
Diagram 3a point H ( )and
write it on the first
Diagram 3a shows a cuboid, line to the right.
ABCDEFGH. Name the angle
between the plane, AGH and the
plane, ABCD

33. ACTIVITY 5 : To Identified The Angle Between Two Planes
G
H
 H A D
E F
D C Bottom
A B
7. In the diagram 3a,
Diagram 3a complete the ∆ HAD
and mark the  HAD
 Angle between the plane, AGH and the plane, ABCD
=  HAD

34. EXAMPLE 3(b)
H 12cm
G
5cm
E F
D C
4cm
A B
Diagram 3b
Diagram 3b shows a cuboid with horizontal
rectangle base ABCD. Name the angle
between the plane ACH and the plane CDHG

35. THANK YOU