Second, the determination method of parallel lines

Third, the nature of parallel lines.

Fourth, translation.

1, translation: the parallel motion of a graph is translation.

2. Two elements of translation:

(1) direction; ② Distance

3. Translation function:

(1) The shape and size of the graph remain unchanged;

(2) The connecting lines of the corresponding points are parallel and equal (both are translation distances)

[Typical example]

1, (1) The straight line AB and the point P are known. If the straight line passing through point P is parallel to AB, then such a straight line ().

A. there is one and only one

B.there are two

C. does not exist

D. does not exist or only has one

Analysis: At present, only one parallel line passes through a point outside the straight line to make a known straight line within the Euclidean geometry; However, if a point is on a known straight line, it is impossible to draw a parallel line through it.

So a is wrong, ignoring a point can be on a known straight line. B error, you can't draw two. C error, ignoring a point outside the straight line. correct

(2) If four straight lines in the same plane satisfy a ⊥ b, b ⊥ c and c ⊥ d, the following formula holds ().

a . a∑d

B.b⊥d

C.a⊥d

d . b∑c

Analysis: Obviously D is wrong. "In the same plane, two straight lines perpendicular to the same straight line are parallel to each other", from which we can get: a∨c, b∨d, and then a⊥b, which can be explained by the nature of parallel lines. So, A and B are both wrong. C is correct.

(3) The following statement is wrong ()

A. Translation will not change the shape and size of the graph.

The translation distance of each point on the graph can be different.

C. After translation, the line segment corresponding to the figure and the corresponding angle are equal respectively.

D, after translation, the connecting line segments of the corresponding points of the graph are equal.

Analysis: Examining the basic concepts of translation, A, C and D are all correct understandings. Option b, "the distance of each point on the graph moving in translation", is actually the length of the connecting line segment of the corresponding point before and after the graph translation, so it should be the same. So choose B for this question.

2. As shown in the figure, AB∨CD, ∠ 1=∠2 is known and verified as EF∨GH.

Analysis: To prove EF∨GH, just prove ∠MFE=∠FHG.

Proof: AB∨CD

∴∠MFA =∞∠FHC (two straight lines are parallel with the same angle)

∵∠ 1=∠2

∴∠MFA+∠ 1=∠FHC+∠2，

That is, ∠MFE=∠FHG.

∴EF∥GH (same angle, two parallel straight lines)

3. As shown in the figure, EF∨AD, ∠ 1=∠2, ∠BAC=700, find the degree of ∠AGD.

Analysis: To find the degree of ∠AGD, you only need to prove GD∨AB, and then you can know that ∠AGD and ∠BAC are complementary by "two straight lines are parallel and the internal angles on the same side are complementary", so it is easy to find the degree of ∠AGD.

Solution: EF∑AD,

∴∠ 2 =∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠

∫≈ 1 =∠2

∴∠ 1=∠3

∴GD∥AB (internal dislocation angles are equal and two straight lines are parallel)

∴∠ BAC +∠∠ AGD =1800 (complementary angles of two parallel lines with internal angles)

∫∠BAC = 700

∴∠agd= 1800—700= 1 100

4. As shown in the figure, it is known that DEC, ∠DEC:∠ECB=2: 1, DC divides ∠ECB equally, and finds the number of times ∠ D.

Analysis: Because DEC needs ∠D and only ∠ 1, and ∠ 1 is half of ∠ECB, DEC knows ∠ DEC+∠ECB = 65438.

Solution: ∫DE∨BC

∴∠

∠DEC+∠ECB= 1800 (two parallel lines are complementary angles of each other)

∠∠DEC:∠ECB = 2: 1。

∴∠ECB=600

∫DC split ∠ ECB

∴∠ 1= 1/2∠ECB=300

∴∠D=300

5. As shown in figure AB∑ED, explain the quantitative relationship between ∠ 1, ∠2 and ∠BCD.

Analysis: by adding auxiliary lines, the angle is transformed from parallel relationship, and then the relationship between triangles is explored.

Solution: ∠BCD+∠2—∠ 1= 1800 The reasons are as follows:

As shown in the figure, a straight line CF∨AB passes through point C.

∫CF∨AB

∴∠ 3 =∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠

∫CF∨AB，AB∨ED

∴CF∥ED (two lines parallel to the same line are parallel to each other)

∴∠4+∠2= 1800 (complementary angles of two parallel lines with internal angles) ②

①+② Available:

∠3+∠4+∠2=∠ 1+ 1800

Namely: ∠BCD+∠2=∠ 1+ 1800.

∴∠BCD+∠2—∠ 1= 1800

6. As shown in the figure, translate △ABC to move point A to point A', and draw the translated △A'B'C'

Analysis: After the graphic is translated, the line segments connected by the corresponding points are parallel and equal. Connecting A A', according to the direction and length of line segment A A', it is easy to make corresponding points B' and C' of point B and point C, thus determining △A'B'C'

Solution: As shown in the figure, connect AA', cross the parallel L with point B as AA', and intercept BB'= AA' on L, then point B is the corresponding point of point B. 。

Similarly, the C' point can also be determined.

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