4. Mutual Positions of Two Planes

Rule: In space, we can find three different mutual positions for two planes:

1. Coincident planes.

2. Intersecting planes.

3. Parallel planes.

Theorem:

1. If a plane passes through a line parallel to another plane and intersects that plane then the line of intersection of two planes is parallel to the given line.

2. If two intersecting lines in a plane are respectively parallel to two intersecting lines in another plane, the planes are parallel.

3. Through a point not in a plane there can be drawn one and only one plane parallel to the given plane.

4. If a line intersects one of two parallel planes, it intersects the other too.

Proofs:

1 . Let d be a line parallel to a plane α, and

β be a plane containing d and intersecting α along line m (Figure 1.20).

Then, d and m lie in β.

Since m is in α and d // α, d and m can not intersect each other.

Therefore they are parallel.

2 . Let α and β be two planes.

Let m, n be two intersecting lines in

α and let m₁, n₁ be two intersecting lines in β so that m // m₁ and n // n₁(Figure 1.23).

We need to prove that α and β are parallel, namely they do not have any common point.

Assume that they have a common point.

Then they will have a common line. Let d be this line.

Since m and n are parallel to m₁ and n₁ respectively both m and n are parallel to β.

So none of m and n can intersect line d.

Since m, n and d are in the same plane, m // d and n // d.

But in this case, m and n must be coincident or parallel lines.

However it is given that they are intersecting lines. So there is a contradiction.

Hence α and β can not have any common point. That means they are parallel planes.

3. Let α be a plane and A be a point not in α.

We need to prove that

a. Through A, there can be drawn a plane parallel to α.

b. This plane is unique.

a. The plane exists

L et d and m be two intersecting lines in α.

Through A there can be drawn a line parallel to d and another line parallel to m.

Let us name these lines as d₁ and m₁.

Lines d₁ and m₁ are intersecting lines. So they determine a plane β (Figure 1.24).

By previous theorem α and β are parallel.

b. The plane is unique

Assume that there is another plane β', containing A and parallel to α.

β' can not contain both d₁ and m₁.

So at least one of d₁ and m₁ intersects β'. Let d₁ be this line.

Since d and d₁ are parallel, d also intersects plane β'.

This contradicts with the parallelity of β' and α.

So plane β is unique.

4. Let α and β be two parallel planes and d be a line intersecting α at a point A (Figure1.25).

We need to prove that d intersects β.

Line d can not lie in β, because d intersects α, and

α // β.

Any line drawn through A and parallel to β must lie in α.

So if d is parallel to β it lies in α.

However we know that d is not in α.

Hence there is only one possibility: Line d intersects β.

Conclusions:

1. The lines parallel to a given plane and passing through a given point not in the given plane lie in the plane parallel to the given plane, and containing the given point.

2. Through a line parallel to a given plane there can be drawn a unique plane parallel to the given plane.

3. Any line in any of two parallel planes is parallel to the other plane.

4. If a line is parallel to one of two parallel planes it is whether in the second plane or parallel to the second plane.

Example 15: Prove that if two parallel lines lie in two different intersecting planes, the intersection of the planes is parallel to the given lines.

Solution: Let d and k be two parallel lines lying in two intersecting planes α and β, respectively (Figure 1.21).

Let m be the intersection of α and β.

Lines d and m are in the same plane.

Since d is parallel to one line lying in β (line k) it is parallel to plane β.

Since m is in β, d can not intersect m.

So d and m are parallel.

By analogy, k and m are parallel too.

Example 16: Prove that, if a line is parallel to two intersecting planes, it is parallel to the intersection of these planes.

Solution: Let α and β be two planes intersecting along line d and m be a

line parallel to both α and β. (Figure 1.22)

Since m is parallel to α and β, in α and β there can be found lines n and k parallel to m.

Since n // m and k // m, n // k .

Then by example 13, d // n and d // k.

Since d // n and n // m, d // m.

Example 17: Show that any line in any of two parallel planes is parallel to the other plane.

S olution: Let α and β be two parallel planes and m be a line lying in α.

m and β can not have any common point because in this case that point will be a common point of α and β which contradicts with the parallelity of α and β.

Therefore, it can be concluded that m // β.

Example 18: Show that there can be drawn a unique plane containing one of

two given skew lines and parallel to the other line.

S olution: In Example 12, it is proven that there can be drawn a plane containing one of two skew lines and parallel to the other line.

So we need to prove that this plane is

unique.

Let m and d be two skew lines.

Assume that there can be drawn two planes α and β containing d and parallel to m.

But in this case, according to Example 14, m should be parallel to d.

This is a contradiction, because m and d are skew lines.

Therefore there can not be drawn more than one plane satisfying the given conditions.

Check Yourself 5

1. α and β are two intersecting planes and A is a point not in α or β.

Show that through A there can be drawn one and only one line parallel to both α and β.

2. α and β are two intersecting planes and m is a line intersecting α and β at distinct points.

a) Show that through m there can be drawn a plane λ whose intersections with planes α and β are parallel.

b) Show that λ is unique.

3. From a point A outside of a plane P we draw parallel lines to plane P. What is the union of all these parallel lines?

4. Which of the following is/are incorrect?

i. If a line is perpendicular to one of two parallel planes it must be parallel to other plane.

ii. If a line intersects one of two intersected planes it must be intersected by second plane also.

iii. If a line is perpendicular to one of two parallel planes it must be perpendicular to second plane.

iv. If two planes are parallel their intersection is a point.

5. Two planes in space are intersected.

a) Can we draw a line that intersects one of the planes but does not intersect the other plane?

b) Can we draw a line that intersects both of the planes?

c) Can we draw a line that does not intersect any of the planes?

Explain and draw your answer.

6. Two skew lines are given in space. Can we say that all planes in that space must intersect at least one of these lines?

Answers

1. Hint: Look at Example 14.

2. Hint: Use theorems 1 and 2 in mutual positions of two planes.

3. A plane parallel to plane P.

4. i,ii and iv

5. a) Yes, take a point A on one of the planes and draw a parallel line to other plane from A.

b) Take two points A and B from different planes and connect them.

c) Take a point A outside of these planes and draw a parallel line that is parallel to both of the planes.

6. No, the plane may be parallel to both of the lines.