First, multiple choice questions
Three straight lines in (1) space intersect in pairs, so the number of planes can be determined by them as follows.
[ ]
A. 1
C. 1 or 3D
(2) Out-of-plane straight lines A and B are in two planes α and β, respectively. If α ∩ β = straight line c, then c []
A. intersects with both a and B.
B. intersect with at most one of a and b.
C. intersect with at least one of a and b.
D. does not intersect with a and B.
(3) Give the following four propositions.
③ If a‖b and a‖α, then b‖α.
(4) if a‖α and b‖α, then a ‖ b.
(A, B, L are straight lines, and α is a plane)
Among them, the number of wrong propositions is []
A. 1
C.3 D.4
(4) Give the following three propositions.
A: The intersection lines of L and M are both within α, and neither is within β.
B: At least one of L and M intersects with β.
C: α intersects β.
When [] is created.
A.b is the necessary and sufficient condition and the unnecessary condition of C.
B.b is a necessary but not sufficient condition for C.
C.b is a necessary and sufficient condition for c.
D.b is not a sufficient or necessary condition for c.
(5) Given straight lines A, B and C and planes α and β, if a⊥α, then []
(6) The projection of two straight lines on different planes in one plane must be []
A. Two intersecting straight lines
B. Two parallel straight lines
C. a straight line and a point outside the straight line
D. There are all three possibilities.
(7) If there is a straight line A in one face of an acute dihedral angle, there is a straight line perpendicular to A in the other face [].
A. There is only one article B, and there are infinite articles.
There is one or infinite number of c.d., which is uncertain.
(8) In space, the following proposition holds []
Two points outside plane a, and one and only one plane is perpendicular to plane a.
B If the straight line L is perpendicular to countless straight lines in the plane α, then l⊥α
The projection of two parallel lines on a plane must be two parallel lines.
D if the distance from point p to three sides of the triangle is equal, and the projection o of p on the plane of the triangle is within the triangle, then o is the center of the triangle.
Second, fill in the blanks
(9) The straight line AB=5cm, and the distances from A and B to the plane α are 1cm and 1.5cm respectively, so the angle formed by the straight line AB and the plane α is _ _ _ _ _ _.
(10) Given plane α ‖ plane β, if a vertical line segment AB=4 and a diagonal line segment CD=6 are sandwiched between α and β, if AC=BD=3, and the midpoints of AB and CD are m and n respectively, then Mn = _ _ _ _ _ _ (where a, c.b, D∈β)
In (1 1) cube ABCD-a1b1c1d1,if m and n are A 1A and B 1B, respectively.
(12) The angles of the three rays PA, PB and PC passing through a point P in space are all 60, so the tangent function value of the angle formed by the ray PC and the plane APB is _ _ _ _ _.
Third, answer questions.
(13) It is proved that four lines that intersect in space and have no * * * points must be * * planes.
(14) As shown in Figure 21-1,E, F, G, H, M and N are the midpoint of the sides of AB, BC, CD and DA and the diagonals AC and BD respectively. If AB=BC=CD=AD, verify:
(ⅰ)ac⊥bd；
(ii) Surface BMN⊥ Surface EFGH ..
(15) As shown in Figure 2 1-2, ABCD is rhombic, and ∠ ABC = 60, PD⊥ surface ABCD, and PD=a, e is the midpoint of PB.
Verify aec⊥ABCD；; ;
(ii) Find the distance from E to the surface pad;
(iii) Find the tangent function of dihedral angle B-AE-C. 。
Answers and tips
One,
( 1)C(2)C(3)D(4)C(5)C(6)D(7)B(8)D
point out
(3) None of the four propositions are correct.
①l may intersect with α; ②l may intersect α, but its intersection point is not on A and B; ③b may be within α; ④a and B can intersect or be different.
(4) When B holds, α must intersect β; On the other hand, when C holds, at least one of L and M intersects β, otherwise l//m contradicts A. 。
(7) The straight line perpendicular to the projection of A on another plane must also be perpendicular to A, so there are infinite straight lines.
(8)(A) When a straight line passing through two points is ⊥ α, all planes passing through the straight line are ⊥α;
(b) When L is an oblique line of α, the straight line perpendicular to the projection of L in α must also be perpendicular to L;
(c) It can be a straight line, two intersecting straight lines, two parallel lines or a point outside a straight line and a line;
Correct.
Three. (13) As shown in answer 2 1- 1, it is known that four straight lines A, B, C and D intersect in pairs, but there are no * * * points. Let a∩b=A, then plane α can be determined by a and b, then we might as well set c ∩ a.
Pairs of C and D intersect without * * * points, which does not exclude points A, B, C * * and D, but there is always a point between C and D.
( 14)(ⅰ)∵ab = ad，BN=ND，∴AN⊥BD
(2) by (1) BD ⊥ Mn. Er //BD, ∴BD⊥EH.
Similarly, MN⊥EF
∴MN⊥ face EFGH
(15) (1) Answer 2 1-2 As shown in the figure, AC and BD intersect at 0, ∫E is the midpoint of PA, and O is the midpoint of AC.
∴EO//PC and ∵PC⊥ ABCD
∴ surface BED⊥ surface ABCD
(ii) * Ethics Office//Personal Computer, ∴EO// People's Bank
The distance from e to PBC is the distance from o to PBC.
You ∵PC⊥ face ABCD, face PBC⊥ face ABCD.
If O is used as OH⊥BC in H, then OH⊥ surface PBC.
(3) BDE⊥ BDE, AO⊥BD BDE, AO ⊥ BDE.
If a is AF⊥BE in f, then OF⊥BE.
Then ∠AFO is the plane angle of dihedral angle A-Be-D.