1. Angle Between Lines and Planes

2. Identifying Planes A plane is a flat surface. Definition Examples:B C D E F G H
ABCD
BCGF
CGHD
BFEA
EFGH

3. Lines on a Plane Identify The line AC lies on the plane ABCD. E H F G

4. Lines on a Plane Identify The line AH lies on the plane ADHE. E H F GB C

5. Lines on a Plane Identify The line AG intersects with the plane EFCD.B C

6. Normals to a Plane
A line which is perpendicular to any line on the plane that passes through the point of intersection of the line with the plane
Definition
Normal
Plane

7. Angle Between Lines and Planes
It is the angle between the line and its orthogonal projection on the plane.
Definition
AOB is the angle between the line OA and the plane PQRS. A P Q S R B O

8. Angle Between Lines and Planes
Name the angle between the line BH and the plane BCGF.
Example 1 A B C D E F G H

9. Angle Between Lines and Planes
Name the angle between the line BH and the plane BCGF.
Example 1
Solution:
The line HG is the normal to the plane BCGF. E H
BG is the orthogonal projection of the line BH on the plane BCGF. F G A D
 HBG is the angle between the BH and the plane BCGF. B C

10. Angle Between Lines and Planes
Name the angle between the line BH and the plane EFGH.
Example 2 E H F G A D B C

11. Angle Between Lines and Planes
Name the angle between the line BH and the plane EFGH.
Example 2
Solution:
The line BF is the normal to the plane EFGH. E H
FH is the orthogonal projection of the line BH on the plane EFGH. F G A D
 BHF is the angle between the BH and the plane EFGH. B C

12. Angle Between Lines and Planes
Name the angle between the line BH and the plane ABFE.
Example 3 E H F G A D B C

13. Angle Between Lines and Planes
Name the angle between the line BH and the plane ABFE.
Example 3
Solution:
The line EH is the normal to the plane ABFE. E H
BE is the orthogonal projection of the line BH on the plane ABFE. F G A D
 HBE is the angle between the BH and the plane ABFE. B C

14. Angle Between Lines and Planes
The diagram below shows a model of a cuboid which is made of iron rods. Calculate
Example 4
the length CE,
the angle between the line CE and the plane BCGF. A B C D E F G H
12 cm
10 cm
8 cm

15. Angle Between Lines and Planes
Example 4 B C F
12 cm
10 cm
Solution: F C E
8 cm
(a) A B C D E F G H
In ∆BCF,
CF2 = BC2 + BF2
8 cm
Pythagoras’ theorem =
10 cm
In ∆FCE,
CE2 = CF2 + FE2
Pythagoras’ theorem
12 cm =
= 308
CE = cm

16. Angle Between Lines and Planes
Example 4
Solution: F C E
8 cm
(b) A B C D E F G H
12 cm
10 cm
8 cm
The angle between the line CE and the plane BCGF is ECF.
In ∆FCE,
sin ECF = =
ECF = 27° 7'

17. The End