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Parallelism between lines and planes

Parallelism in dihedral rests on a simple idea: if two elements are parallel in space, their matching projections are parallel too. The exception is profile lines, which need the third projection because two profile lines can have parallel projections and not be parallel in space.

What you need to know

• Recognise two parallel lines from their matching projections.
• Draw a line through a point parallel to a given one.
• Check whether a line is parallel to a plane.
• Draw the plane parallel to a given one through an outside point.

How it is solved

1. Line parallel to a plane: it is enough that it is parallel to a line contained in the plane.
2. Plane parallel to another through a point: draw through the point a line parallel to one of the plane and complete the plane with a second direction.
3. Two planes given by traces are parallel if their matching traces are parallel.
4. If there are profile lines, check in the third projection before saying they are parallel.

How it appears in the exam

It appears as “draw the plane parallel to α through P” and as a step inside angles and distances. Watch the profile lines: it is the case that slips into exams and the one most often failed.

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