Proof method for determining mathematical similarity in the ninth grade of the Shanghai Science and

What does this title mean?

mathematics or maths, from the Greek, "máthēma"; Often abbreviated as "math"), it is a discipline that studies concepts such as quantity, structure, change, space, and information, and is a kind of formal science from a certain point of view. Mathematicians and philosophers have a range of opinions about the exact scope and definition of mathematics.

Because most of the knowledge of triangle similarity will be used to prove that two triangles are similar in the future, it is enough to memorize the following methods to prove triangle similarity, and prove that triangles are similar:

1. A straight line parallel to one side of the triangle intersects with the other two sides (or extension lines on both sides), and the triangle formed is similar to the original triangle;

2. The two angles of a triangle correspond to the two angles of another triangle, so the two triangles are similar.

3. The ratios of the two corresponding sides of the two groups of triangles are equal, and the angles of the corresponding states are equal.

4. The ratios of the three corresponding sides of the two triangles are equal.

5. Two triangles with equal corresponding angles and proportional sides corresponding to the calendar are called similar triangles.

It's not hard. 1. Corresponding to the figure with equal angles and unequal corresponding edges, similar; Figures with equal corresponding angles and equal corresponding edges, congruent.

In the ninth grade, I mainly learned about triangles, and of course, I should also know that all equilateral triangles are similar, and all squares are similar,..i.e. all regular n-sided shapes are similar.

1) Isosceles triangle: isosceles triangles with equal apex angles are similar; Isosceles triangles with equal base angles are similar; Equilateral triangles are similar.

2) Right triangle: In addition to right angles, there is an angle that corresponds to an equal right triangle; Isosceles right triangles are similar.

3) Arbitrary triangle: There are two angles corresponding to the similarity of equal triangles.

4) The inner angles of similar triangles correspond to equals.

5) Congruent triangles are special similar triangles, and their corresponding angles are equal and their corresponding sides are equal.

2. A triangle with two sides corresponding to a proportional triangle is a similar triangle, such as a a'=b/b'

1.Proof: Because there is a common right angle c. So only the other angle needs to be equal to prove similarity. According to known conditions. b=30°。Angular DAC also = 30°. So it was proven.

Analysis: Tell the four line segments (that is, the two groups of corresponding edges are equal, as long as the third group of corresponding edges is also equal or the angles are equal, and find that the angles are a group of pairs of top angles, so they are equal, so the idea is clear.

Proof: Because oa = quadruple root number three, oc = double root number three, so oa oc = 2, ob od = 4 2 = 2, so oa oc = ob od, and because aoc = cod so: aob cod.

ABC vs. A'b'c'It's a bit-like graph, so, abc a'b'c'

1）∵△abc∽△a'b'c'，∴∠cab=∠c'a'b', if the isotope angle is equal, the two lines are parallel, so ac a'c'

2）∵△abc∽△a'b'c'，∴ac/a'c'=ab/a'b'=2, which yields ac=2a'c', certified ACA'c', hence a'c'is the median line of the OAC. So cc'=oc'=5.

Proof: ABC vs. A'b'c'It's like the figure ABC A'b'c'

a=∠c'a'o

ac‖a'c'

Solution: ABC A'b'c'

abc∽△a'b'c'

ac:a'c'=ab:a'b'=2

a'c'=½ac

a'c'is the median line of the ACO.

c'is the midpoint of the OC.

oc=oc'=5

Open the math book and follow those parallel theorems.

(1) Because the triangle ABC and triangles A, B, C, are similar figures, these two triangles are similar triangles.

So the angle ACB is equal to a, c, b,

So ac is parallel to a, c,

2) Because ab is known to be equal to 2a, b, .

So the similarity ratio of the two triangles is 2 to 1

So they also seem to have a 2-to-1 ratio

So OC is equal to 2 times OC, which is equal to 10

Then cc, is equal to oc minus oc, equal to 5.

Additional note: ", the meaning of the representative is "apostrophe". (1) Ask the proof that you might think of straight lines ac and a, c

It is not truncated by the straight line cb, and it cannot be proved parallel by the method of isotopic angle, in fact, it is only necessary to extend the two straight lines of cb and a, c, and so on.

I think it's better to study congruent triangles thoroughly and then compare similar triangles.

Hope it helps.

In fact, it is very simple, similar shapes have two characteristics: (1) the angles are equal and (2) the sides are proportional.

In abc, abc=60°, the point p is a point within abc, such that apb= bpc= cpa, and pa=8, pc=6, then pb=

abc=∠abp+∠cbp=60°

BPC=120° is known, so PBC+ PCB=60°= ABP+ CBP

So pcb= abp

The same can be said for PBC = BAP

So the triangular PAB and PBC are similar.

pa:pb=pb:pc

pb = PA*pc = 4 under the root number 3

How to determine similar triangles.

Judging by the characteristics of similar figures. (The corresponding sides are proportional, and the angles of the corresponding edges are equal).

Method 1: A straight line parallel to one side of the triangle intersects with the other two sides (or extension lines on both sides), and the triangle formed is similar to the original triangle; (This is the lemma of the similarity triangle judgment, which is the basis for the following determination method proof.) The proof method of this lemma requires the proof that the parallel lines divide the line segments proportionally) Method 2.

If the two angles of one triangle correspond equally to the two angles of another triangle, then the two triangles are similar; (aa) Method III.

If the ratios of the two sets of corresponding sides of two triangles are equal, and the corresponding angles are equal, then the two triangles are similar; (SAS) Method IV.

If the ratios of the three sets of corresponding sides of two triangles are equal, then the two triangles are similar; (SSS) Method 5.

Two triangles with equal corresponding angles and proportional sides are called similar triangles (proven by definition).

The corners correspond to equal.

The edges of the strips correspond to each other, and the angles are equal.

The edges correspond proportionally.

4. In a right-angled triangle, the hypotenuse and a right-angled side correspond to the proportional 5, and the parallel (line) is similar.

There are 2 angles equal.

Either the 3-sided ratios correspond to the same or the 2-sided ratios are equal and the angles on both sides are equal.

, the three sides correspond proportionally.

The two sides correspond proportionally, and the angle is the same.

The two angles correspond to equal.

There is a theorem for similar triangles.