Determination of congruent triangles in Mathematics for the second year of junior high school

2) Because the triangle AEB is all equal to the triangle EBC

So ae = ec = 1 ac of 2

And because ac=bf

So EC = 1/2bf

Triangle is the basic geometric shape, in the primary, junior high school, high school textbooks have about the calculation of triangles, the determination of congruent triangles is a common test in the high school entrance examination, this will involve fill-in-the-blank questions, solution questions and so on. Only two triangles that coincide exactly are considered congruent triangles. Then, when arguing a congruent triangle, it is necessary to argue from the perspective of the triangle and the length of the sides.

1. Edge-by-edge (SSS).

The edge-edge theorem, or SSS for short, is one of the important theorems in plane geometry. The theorem is that there are three sides corresponding to two equal triangle congruences. It is used to prove the congruence of two triangles. This theorem was first proved by Euclid.

2. Corner edge (SAS).

If the length of two sides of each triangle is equal, and the angles between the two sides (i.e., the angles formed by the two sides) are equal, the two triangles are congruent triangles.

3. Corner Corner (ASA).

The two corners and their edges correspond to two equal triangle congruences, abbreviated as "corner corners" or "asa".

Corner corner is one of the methods of determining the congruence of triangle, it should be noted that the edge in the corner corner must be an edge common to both corners (a corner is composed of two sides, and any two corners in the triangle have a common edge).

Fourth, the corner edge (AAS).

A corner corner is a common edge of two corners and these two corners, and the corner edge theorem can deduce congruence. A corner edge is two corners and another non-common edge, and the corner edge can also be congruent.

5. Right angle edge (HL).

The HL theorem is a theorem that proves the congruence of two right triangles by proving that the right sides and hypotenuse of two right triangles correspond to the congruence.

The decision theorem is that if the hypotenuse and one right-angled side of two right triangles correspond to the same, then the congruence of the two right triangles (abbreviated as hl) is a special determination method that can be converted to ASA

aaa (angle-angle-angle): Triangles are equal, and they cannot be congruent, but they can prove similar triangles.

SSA (Side-Side-Angle): One of the corners is equal, and the two sides of the non-included angle are equal.

To prove AAS, the condition should be a= a, b= b, bc=b c or ac=a c, not ab between two corners, otherwise it is asa

Of course, known angles can also be other angles, but if you want to use AAS, you just don't choose the edge.

In its entirety, there should be two triangles with two corners and the opposite side of one of them corresponding to two equal congruences.

The opposite side of a corner is an edge that is not the edge of the corner. It's so awkward.

For example, in the triangle ABC, the opposite side of the angle A is BC.

That's it...

It can also be said that except for the edge of the two corners, the other two sides are considered "the opposite side of one of the corners".

Upstairs is right, a triangle has three sides, three corners, and the opposite side of one corner is the sandwich of the two sides adjacent to this corner. For example, the right angle of a right triangle is the hypotenuse.

This refers to the edge of a triangle where either of the two corners is opposed.

1. Three groups of two triangles with equal sides (SSS or "edge-edge-edge") also explains the reason for the stability of triangles.

2. There are two triangles with equal sides and their angles corresponding to the congruence (SAS or "corner edges").

3. There are two corners and their sandwich edges corresponding to two equal triangles congruence (ASA or "corner corners").

4. There are two corners and the opposite side of one of the corners corresponds to two equal triangles congruence (AAS or "corner edges").

5. The congruence conditions of a right triangle are: the hypotenuse and the straight angle side correspond to the equal congruence of two right triangles (hl or "hypotenuse, right angled side").

Therefore, sss, sas, asa, aas, and hl are all theorems that determine the congruence of triangles.

Note: In the congruence determination, there are no AAA angles and SSA (exception: the right triangle is HL, which belongs to SSA) side corners, neither of which can uniquely determine the shape of the triangle.

6.The three middle lines (or high and angular divisions) correspond to two equal triangle congruences.

[1] Angular AOB = EOF

So aob-eob=eof-eob

i.e. aoe=bof

Because it is an rt triangle, ao=bo, eo=fo in aoe and bof, ao=, eo=fo, so aoe is all equal to bof

So ae=bf

Extend AE to cross BF at point G

I forgot that I was a freshman in high school, and I rarely did geometry Su Xue, who was in my first year of high school, was annoyed.

Because the angle aoe angle eob=90 degrees.

Angle bof + angle boe = 90 degrees.

So angular aoe = angular bof

Because of the isosceles triangle boa and eof

So oe=of oa=ob

So the triangle aoe is all equal to the triangle bof

So ae=bf

Because the triangle aoe is all equal to the triangle bof

So the angle obf = the angle aoe

Because the angle aoe eab + abo = 90 degrees.

So the angle obf eab+abo=90 degrees.

So the triangle is abm.

Angle AMB 90 degrees.

So AE is perpendicular to BF

Because both CBF and ACE are redundant with ECB.

So ace= fbc

Because bc=ac, ace is all equal to bcf, so cf=ae

So bf = ce = cf + ef = ae + ef

So ef=bf-ae

Because d is any point on ab, when d is close to b, bf ae so ef=| bf-ae |