If line A is parallel to plane α, what is the positional relationship between line A and the line in plane α?
Parallel or different planes.
If line A is parallel to plane α, how many lines in plane α are parallel to line A? What is the positional relationship between these straight lines?
A: countless articles; Parallel.
If line A is parallel to plane α, and plane β passing through line A intersects plane α at line B, what is the positional relationship between lines A and B? Why?
Parallel; Because A∧α, A and α have no common point, A and B have no common point, and A and B are on the same plane β, so A and B are parallel.
To sum up, what conclusion can we draw under the condition that the straight line A is parallel to the plane α?
If a straight line is parallel to a plane, then the intersection of any plane passing through this straight line and this plane is parallel to this straight line.
Exercise questions:
1. (Questioning rammed foundation) Judge whether the following conclusions are true or false. (Mark "√" correctly and "×" incorrectly. )
(1) If a straight line is parallel to a straight line in a plane, then the straight line is parallel to the plane. ()
(2) If the straight line a∑ plane α, P∈α, there are countless straight lines passing through point P and parallel to straight line A. ()
(3) If two straight lines in one plane are parallel to the other plane, the two planes are parallel. ()
(4) If two planes are parallel, then two straight lines in these two planes are parallel or different planes. ()
Answer: (1)×(2)×(3)×(4)√.
2. The following proposition is correct ()
A. if a and b are two straight lines, A∑B, then a is parallel to any plane β passing through b.
B If line A and plane α satisfy a∧α, then A is parallel to any line within α.
Two planes parallel to the same line are parallel.
D if lines a and b and plane α satisfy a∨b, a∧α, b? Alpha, then b ∧ alpha
Analysis: in option a, is a∧β or a? β, A is incorrect.
In option B, the straight lines in A and α are parallel or different, and B is wrong.
Two planes in C are parallel or intersect, and C is incorrect.
From the perspective of parallelism between lines and faces, option D is correct.
Answer: d
3. Let α and β be two different planes, m is a straight line, m? α. "m ∨ β" is () of "α ∧β"
A. Sufficient and unnecessary conditions
B. Necessary but not sufficient conditions
C. Sufficient and necessary conditions
D. Conditions that are neither sufficient nor necessary
Analysis: by m? α，m∧βα∧β。
But m? α,α∥β? m∧β，
∴ "m ∧ β" is a necessary and sufficient condition for "α ∧β".
Answer: b