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Angles between lines and planes

An angle can only be measured in true size. In dihedral that means rotating, spinning or changing the plane until the two elements forming the angle are parallel to the projection plane. Never measure on a foreshortened projection: the number you get is not the one in the statement.

What you need to know

• Angle between two intersecting lines: rotate the plane that contains them flat.
• Angle between two skew lines: move a parallel until it cuts the other.
• Angle between line and plane: angle between the line and its projection on the plane.
• Angle between two planes: angle between their steepest lines, or between their normals.

How it is solved

1. Identify the two elements and find a plane that contains them or is parallel to both.
2. Rotate that plane flat or change the plane until you see it in true size.
3. Measure the angle on the rotation, not on the original projections.
4. Check the order of magnitude: an angle between planes never goes above 90°.

How it appears in the exam

“Angle between line r and plane α” and “angle between two planes” are the two typical statements. In the second one the shortcut is to use the normals: the angle between planes is the same as the one between their perpendiculars.

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