Parallel lines and planes in space

1. Apply a parallelism criterion with all its incidence and outside-plane conditions.

2. Calculate or simplify this relation.

   $$l\parallel m\subseteq\alpha,\ l\not\subseteq\alpha\Rightarrow l\parallel\alpha$$

3. Without the outside-plane condition, l could be contained in α.

l is parallel to α.

Checks and common pitfalls: Without the outside-plane condition, l could be contained in α.

1. Apply a parallelism criterion with all its incidence and outside-plane conditions.

2. Calculate or simplify this relation.

   $$EF\parallel BC,\quad BC\subseteq(BCD)$$

3. The tetrahedron is nondegenerate, so E and F do not lie in the base plane.

EF∥BC and EF is outside plane BCD, hence EF∥plane BCD.

Checks and common pitfalls: The tetrahedron is nondegenerate, so E and F do not lie in the base plane.

1. Apply a parallelism criterion with all its incidence and outside-plane conditions.

2. Calculate or simplify this relation.

   $$d=|b-a|=|2-0|=2$$

3. An oblique joining segment would be longer than the plane distance.

The requested value is 2.

Checks and common pitfalls: An oblique joining segment would be longer than the plane distance.

1. Find an explicit parallel line inside the top plane.

2. Calculate or simplify this relation.

   $$m=(t,0,4)\subseteq\alpha;\quad l\parallel m$$

3. Both the direction and outside-plane condition are verified.

It does not meet the top plane and is parallel to the corresponding top edge.

Checks and common pitfalls: Both the direction and outside-plane condition are verified.

1. Apply the midpoint theorem in two faces.

2. Calculate or simplify this relation.

   $$EF\parallel BC,\quad EG\parallel BD,\quad EF\cap EG=\{E\}$$

3. One midpoint segment alone would establish only one direction.

EF∥BC and EG∥BD give two intersecting directions.

Checks and common pitfalls: One midpoint segment alone would establish only one direction.

1. Apply a parallelism criterion with all its incidence and outside-plane conditions.

2. Calculate or simplify this relation.

   $$\alpha\parallel\beta,\quad l=\alpha\cap\gamma,\ m=\beta\cap\gamma\Rightarrow l\parallel m$$

3. The contradiction excludes intersection inside the common cutting plane.

They are parallel.

Checks and common pitfalls: The contradiction excludes intersection inside the common cutting plane.

1. Apply a parallelism criterion with all its incidence and outside-plane conditions.

2. Calculate or simplify this relation.

   $$l=(t,0,1),\quad m=(0,s,1),\quad\alpha:z=0$$

3. A shared plane-parallel relation does not determine their relative directions.

No: x- and y-directed lines in z=1 may intersect while both are parallel to z=0.

Checks and common pitfalls: A shared plane-parallel relation does not determine their relative directions.

1. Apply a parallelism criterion with all its incidence and outside-plane conditions.

2. Calculate or simplify this relation.

   $$l,m\subseteq\alpha,\ l\cap m\ne\varnothing,\ l\parallel\beta,\ m\parallel\beta\Rightarrow\alpha\parallel\beta$$

3. Two independent directions constrain the plane orientation.

Two intersecting lines in one plane must each be parallel to the other plane.

Checks and common pitfalls: Two independent directions constrain the plane orientation.

1. Apply a parallelism criterion with all its incidence and outside-plane conditions.

2. Calculate or simplify this relation.

   $$\alpha:z=y,\quad\beta:z=0;\quad l=(t,1,1),\ m=(t,2,2)$$

3. Both lines avoid β but their plane intersects β along y=z=0.

They supply only one direction; α can tilt about that direction and intersect β.

Checks and common pitfalls: Both lines avoid β but their plane intersects β along y=z=0.

1. Apply a parallelism criterion with all its incidence and outside-plane conditions.

2. Calculate or simplify this relation.

   $$d=|b-a|=|3-0|=3$$

3. An oblique joining segment would be longer than the plane distance.

The requested value is 3.

Checks and common pitfalls: An oblique joining segment would be longer than the plane distance.

1. Use horizontal planes and a horizontal line at height c.

2. Calculate or simplify this relation.

   $$\alpha:z=0,\quad\beta:z=1;\quad l=(t,0,c)$$

3. A line already contained in β is not called parallel to β under the disjoint definition.

It can lie in β or be parallel to β.

Checks and common pitfalls: A line already contained in β is not called parallel to β under the disjoint definition.

1. Compare direction vectors.

2. Calculate or simplify this relation.

   $$l\text{ direction }(1,0,0);\quad m\text{ direction }(0,1,0)$$

3. There are many different directions in a plane.

No: l=(t,0,1) is parallel to z=0, but not parallel to the y-axis in that plane.

Checks and common pitfalls: There are many different directions in a plane.

1. Use similar triangles along corresponding rays from the apex.

2. Calculate or simplify this relation.

   $$\lambda=2/3;\quad S_{\text{section}}/S_{\text{base}}=\lambda^2=4/9$$

3. Measuring from the base would give a different linear ratio.

The requested value is 0.44444444.

Checks and common pitfalls: Measuring from the base would give a different linear ratio.