Chaper 6 Quest: ways to determine a plane, theorems, postulates, and properties relating parallel lines and planes

Question 0

Properties relating parallel lines and planes

Answer 0

1. if 2 planes are perp. to the same line, they are ll to each other
2. If a line is perp. to 1 of the 2 ll planes, it is perp. to the other plane as well.
3. If 2 planes are ll to the same plane, they are parallel to each other
4. If 2 lines are perp. to the same plane, they are parallel to each other
5. If a plane is perp. to 1 of the 2 ll lines, it is perp to the other line as well.

Question 1

Ways to determine a plane (#)?

Answer 1

1. 3 non-collinear points
2. A line and a point not on the line
3. 2 intersecting lines
4. 2 ll lines

Question 2

Postulate

Answer 2

If a line intersects a plane not containing it, then the intersection is exactly one point (foot).

Question 3

Postulate 2

Answer 3

If two planes intersect, their intersection is exactly one line.

Question 4

Foot

Answer 4

The point of intersection of a plane and a line NOT on that plane.

Question 5

Theorem 1

Answer 5

If a line is perp. to 2 distinct lines in a plane which pass through its foot, then it is perp. to the plane.

Question 6

Theorem 2

Answer 6

If a line is perp. to a plane, it is perp. to every line in the plane that passes through its foot.

Question 7

Parallel

Answer 7

A line and a plane are parallel if they do not intersect.

Question 8

Parallel planes

Answer 8

2 planes are parallel if they do not intersect.

Question 9

Skew lines

Answer 9

2 lines that are not coplanar (NOTE: 2 parallel lines are ALWAYS coplanar!)

Question 10

Theorem 3

Answer 10

If a plane intersects 2 ll planes, the lines of intersection are ll.

Question 11

Useful proof fragment

Answer 11

S R
AD and BC intersect at E Given
ABCD is a plane 2 intersecting lines determine a plane
m ll n Given
AB ll CD If a plane intersects 2 ll planes, then the lines of intersection are ll