5. Mutual Positions of Three Planes

Rule: In space three planes may have different positions as follows:

1. All of them are parallel to each other.

2. Two of them are coincident and third one is parallel to them.

3. All of them are coincident.

4. Two planes are parallel and third one intersects both of them. (Intersect at two lines)

5. They intersect at three different lines.

6. They intersect at one point.

7. They intersect at one line.

a. b. c. d.

e. f. g.

Theorem:

1. The planes parallel to the same plane are parallel.

2. If two parallel planes are intersected by a third plane, the lines of intersection are parallel.

3. The parallel line segments whose end points are on two parallel planes are equal.

4. When three planes intersect each other if two of intersection lines are parallel then the third intersection line is parallel to these two lines.

P roofs:

1. Let α, β, and  be three planes such that α //  and

β // .

We need to prove that α // β (Figure 1.26).

Assume that α and β are not parallel.

Then they have a common line. Let m be this line.

Through m we can draw only one plane parallel to .

This is a contradiction.

Hence α // β.

2 . Let α and β be two parallel planes and  be a plane

intersecting both α and β.

Let m and d be the intersections of α and , and β and

, respectively (Figure 1.27).

Lines m and d are in the same plane.

If m and d have a common point, this point will be a common point of α and β.

However α and β are given as parallel planes.

That means they can not have a common point.

So, m and d can not have a common point.

Hence, they are parallel.

3. Let α and β be two parallel planes.

Let AB and A₁B₁ be two parallel line segments, such

that points A, A₁ in α and B, B₁ are in β (Figure 1.28).

Since AB and A₁B₁ are parallel lines, they determine a plane λ.

A and A₁ are two common points of λ and α.

So the line passing through A and A₁ is the intersection of λ and α.

By the same logic the intersection of λ and β is the line passing through B and B₁.

Since α // β, AA₁ // BB₁ .

Additionally, it is given that AB // A₁B₁.

Therefore, AA₁B₁B is a parallelogram and AB = A₁B₁.

4. Let α, β and λ be three planes, and m, n and d be the intersections of α – β, α – λ and β – λ, respectively.

Assume that m and n are parallel lines.

Let us prove that d // m (Figure 1.29).

Lines d and m are in the same plane.

If d intersects m at a point A then A will be a common point of planes λ and α, because m is in α and d is in λ.

So, A must be on line n.

Then m and n will have a common point.

However m and n are given as parallel lines.

So there cannot be such a point.

Therefore m // d.

Conclusion:

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

Example 19: Prove that if two of intersection lines formed by three intersecting planes intersect each other, the third intersects these lines at the same point.

Solution: Let α, β and λ be three planes intersecting each other.

Let m be the intersection of α and β, n be the intersection of α and λ, d be the intersection of β and λ.

Let m and n intersect each other at a point A.

W e need to prove that d passes through A (Figure 1.30).

Since m is the intersection of α and β, A is in β, and since n is the intersection of α and λ, A is also in λ.

So A is on d which is the intersection of β and λ.

So d intersects m and n at A.

Example 20: Show that if two intersecting lines in a plane α are parallel to a

plane β then α and β are parallel planes.

S olution: Let m and d be two intersecting lines in α parallel to plane β.

Then in β there can be found two lines m' and d' parallel to m and d, respectively (Figure 1.33).

d' and m' can not be coincident lines. Because in this case m and d will be parallel to the same line which implies their parallelity.

If d' // m' then since d // d', d and m' will be parallel.

Then dand m will be parallel to a common line, that is m'. So d // m.

However we know that they are intersecting lines.

So d' and m' are not parallel.

Hence d' and m' are intersecting lines in β.

Therefore α // β.

Example 21: ABC is a triangle and α is a plane. Show that if sides AB and

BC are parallel to α then AC is also parallel to α.

Solution: AB and BC are two intersecting lines in plane ABC.

Since they are parallel to α, planes ABC and α are parallel.

So AC is parallel to α.