Basic properties of plane
Teaching objectives
1, knowledge and ability:
(1) consolidates the basic properties of the plane, namely, four axioms and three inferences.
(2) Can use axioms and inferences to solve problems.
2, process and method:
(1) Experience the process and method of determining a plane in space;
(2) Master the method of proving three-point * * line, three-line * * point and multi-line * * surface by using the basic properties of plane.
3, emotional attitudes and values:
Cultivate students' attitude of careful observation and habit of careful thinking, and improve their aesthetic ability and spatial imagination.
Teaching focus
The three basic properties of a plane are three inferences.
Teaching difficulties
Accurately apply three axioms and inferences to solve problems.
teaching process
First, the problem situation
Question 1: How many planes can three straight lines of a point in space determine? What about three parallel lines in space?
Question 2: How to judge whether the bottom ends of the four legs of the table are on the same plane?
Second, learn new things by reviewing the old ones.
Axiom 1
If two points on a straight line are on the same plane, then all points on this straight line are on this plane.
Axiom 2
If two planes have a common point, then they have other common points, and the set of these common points is a straight line passing through this common point.
Axiom 3
Passing through three points that are not on the same straight line has one and only one plane.
Inference 1
Through a straight line and a point outside this straight line, there is only one plane.
Inference 2
Passing through two intersecting lines, there is one and only one plane.
Inference 3
Passing through two parallel straight lines, there is only one plane.
Axiom 4 (Parallel Axiom) Two straight lines parallel to the same straight line are parallel to each other.
Compare the above axioms and inferences:
Third, the application of mathematics.
Basic training: (1) Known:; Proof: straight AD, BD, CD*** plane.
Proof:-axiom 3 reasoning 1
-axiom 1
It can also be proved that straight AD, BD, CD***
Reflection on solving problems 1 1. The logic should be rigorous.
2. Writing should be standardized
3. The steps to prove the surface of * * *:
(1) Plane —— Axiom 3 and Its Three Inferences
(2) Prove that the straight line "returns" to the plane (the straight line is in the plane, such as)-axiom 1.
(3) draw a conclusion.
Variant 1. If two straight lines intersect, are these three straight lines * * * planes? (oral answer)
Variant 2. Given four points on the surface of space, a plane can be determined by any three points. How many planes can be determined from these four points?
Variant 3. The four line segments are connected end to end in turn. Does the resulting graph have to be a plane graph? (oral answer)
(2) Known straight line meets: verification: straight line
Proof:-axiom 3 reasoning 3
-axiom 1
Straight line * * *
Improve training: know, verify: four straight lines are on the same plane.
Train of thought analysis: consider using straight lines A and B to determine a plane, and then prove that straight lines C and L are on this plane, but it is very difficult. So we can open our minds, consider determining two planes, and then prove that the two planes overlap, and the problem will be solved.
Prove:
Axiom 3 inference 3
Axiom 3 inference 3
-axiom 1
So the planes pass through two intersecting straight lines at the same time, so the planes overlap. -Axiom 3 Inference 2
Straight line * * *
The above method is called the same method.
Outward bound training: As shown in the figure, in the triangle pyramid A-BCD, E and G are the midpoint of BC and AB, F is on CD, H is on AD, and DF: FC = DH: HA = 2: 3; Verification: EF, GH and BD meet at one point. [The idea of permeating the planarization of spatial problems]
Train of thought analysis: train of thought 1: Open the train of thought and consider three planes. First, prove that two straight lines are in one plane and intersect, and then prove that the intersection point is in two planes. According to axiom 2, it is on the only intersection line of two sides-the third straight line, so we can prove the points of three lines.
Proof 1: connection,
Since E and G are the midpoint of BC and AB respectively, since DF:FC=DH:HA=2:3, axiom 4.
* * * plane, from the above knowledge, intersect, let the intersection point be O, then the plane, plane,
So straight lines, so EF, GH and BD intersect at one point.
Idea 2: First prove that straight lines GH and BD intersect at point P, straight lines EF and BD intersect at point Q, then prove that two points P and Q coincide, and then get that EF, GH and BD intersect at point.
ProOF method 2: prompt: the intersection h is ho, make the intersection o, connect of, prove,
Extend GH and EF so that they intersect the straight line BD at points P and Q respectively. From the similarity of triangles, it can be concluded that OP=OQ. So point p and point q coincide.
Linking life: in cubic wood, try to draw the plane of the midpoint P, Q and R on three sides to cut the cross-sectional shape of wood.
Reflection on solving problems +0. The logic should be rigorous.
2. Writing should be standardized
3. The method should be mastered
(1) Prove the * * * plane:
1) Determine the plane-Axiom 3 and its three inferences-Axiom 3 and its three inferences.
2) Prove that the straight line "returns" to the plane (the straight line is in the plane, such as)-axiom 1
3) Draw a conclusion.
(2) the steps to prove the * * * line:
(1) All points on the first plane (such as a plane)-axiom 1.
② Prove that the point is in the second plane (such as plane)-axiom 1.
③ Conclusion 1: All points are at the intersection of two planes.
④ Conclusion 2: All Points * * * Line-Axiom 2
(3) the steps to prove the * * * point:
Proof of 1) on one point-axiom 3 and 3 inferences
2) Prove that the point lies in two planes (such as planes)-axiom 1.
3) Conclusion 1: This point is at the intersection of two planes-Axiom 2.
4) Conclusion 2: Three lines score * * *
Fourth, review summary
This section mainly reviews three axioms and three inferences of plane, and learns how to use axioms and their inferences to solve problems.
V. Homework (see Homework)
Feedback exercise
[1. 2. 1 Basic Properties of Plane (2)]
1, a plane passing through three points on the same line ()
A, only 1 b, only 3 c, countless d, and 0.
2. If three planes of space intersect, the number of their intersecting lines is ()
A, 1 or 2B, 2 or 3C, 1 or 3D, 1 or 2 or 3.
3. There is () in the plane with equal distance between four points in space.
A, 3 or 7 b, 4 or 10 c, 4 or countless d, 7 or countless.
4, four parallel straight lines can be determined at most ()
A, three planes B, four planes C, five planes D and six planes.
5. Four line segments are connected end to end, and the maximum number of planes they can determine is.
6. Give the following four propositions:
(1) If the four-point space is not * * *, there will be no * * line at three o'clock;
② If a point on the straight line L is out of plane, then L is out of plane;
(3) If a straight line,,, and * * and * *, and * * *;
(4) The plane where three straight lines intersect.
The serial number of all correct propositions is.
7. The point P is on the straight line L, and the straight line L is in the plane, which is indicated by the symbol ().
A.B.C.D.8 The following reasoning is wrong ().
A.B.C.D.9 Here are four propositions (where A and B represent points, lines and planes respectively).
1, 2, 3, 4 The serial number of the proposition with correct narrative method and reasoning process is _ _ _ _ _ _ _ _.
10. It is known that A, B and C are not in a straight line, so it is proved that straight lines AB, BC and CA***.
1 1. Verification: If a straight line intersects two parallel lines, then three straight lines are on the same plane.
Known: straight lines,, and,,;
Verification: straight line, * * * surface.
12, in the cube ABCD-a1b1c1d1,
① Can aa1and CC 1 determine a plane? Why?
② Can points B, C 1 D determine a plane? Why?
③ Draw the intersection of plane ACC 1A 1 and plane BC 1D, and the intersection of plane ACD 1 and plane BDC 1.
13, four intersecting straight lines with no * * * points. (Note: There are two cases, as shown in the figure. Try to prove them separately.)