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Looking for high school math basic exercises. More useful.
Line and plane (1)? 6? 1 practice

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.