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## I. Bridging perspective

In a cavalier perspective representation of a solid :

- a figure represented in a plane seen from the front is represented in true size, without changing its shape;
- two parallel lines are represented by two parallel lines ;
- aligned points are represented by aligned points ;
- the middle of a segment is represented by the middle of the drawn segment;
- the visible elements are in solid lines, those which are hidden are in dotted lines;
- a line perpendicular to the frontal plane is represented by a line making an acute angle with the horizontal of the representation support;
- any length on such a line is multiplied by a coefficient less than 1.

**II. Relative positions of lines and planes**

### 1.Rules of impact

**Rules:**

- Through two distinct points there passes a single line;
- Through three non-aligned points A, B, C, there passes a unique plane noted (ABC) ;
- If a plane contains two points A and B, then it contains all the points of the line (AB) ;
- If (d) is a line and A is a point not on (d), there is a unique plane containing (d) and A.

**2. relative positions of two lines**

**Ownership:**

Two lines can be :

- Coplanar: they are located in the same plane (they are secant or parallel)
- Non coplanar: and in this case they have no points in common.

### 3. relative positions of a line and a plane

A line can be :

- Contained in a plane if it passes through two points of the plane;
- Secant to the plane, if it has only one point in common with this plane (see opposite);
- Parallel to the plane if it has no common point with the plane.

**4.relative position of two planes**

**Ownership:**

Two planes are either parallel, if they have no points in common, or secant and in this case their intersection is a straight line (so they have an infinity of intersection points).

**Example of intersecting planes, along the line (UV).**

## III- Parallelism in space

**1.parallelism between lines**

**Properties:**

- If two lines are parallel to a third, then they are parallel to each other.
- If two lines are parallel then any plane that intersects one also intersects the other.

**2. parallelism between two planes**

- If two planes are parallel then any plane parallel to one is also parallel to the other.
- If two secant lines
*(d*) and*(d’)*of the plane (P) are parallel to two secant lines and of the plane (P’) then the two planes (P) and (P’) are parallel. - If two planes (P) and (P’) are parallel, then any plane that intersects one also intersects the other and the intersecting lines
*(d*) and*(d’*) are parallel.

**Example of parallel planes determined by two pairs of intersecting lines.**

**2.parallelism between lines and planes**

**Properties:**

- If two planes are parallel and if a line is parallel to the first plane then it is also parallel to the second.
- If the line
*(d*) is parallel to the plane (P) then any plane containing*(d*) and intersecting (P) intersects it along a line parallel to*(d*). Demonstration - If the line
*(d*) is parallel to a line of the plane (P) then*(d*) is parallel to the plane (P). - If the planes (P) and (P’) are secant along the line and if
*(d*) is a line parallel to the two planes (P) and (P’) then the lines and*(d*) are parallel.

## IV. Calculations in space geometry

### 1.orthogonality between line and plane

**Ownership:**

- A line is perpendicular to a plane if and only if it is orthogonal to two intersecting lines in that plane.
- If a line is perpendicular to a plane then it is orthogonal to all lines in that plane.

### 2.areas and volumes of classical solids

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