Methods for proving congruent triangles in junior high school mathematics

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").

SSS, SAS, ASA, AAS, and HL are all theorems that determine the congruence of triangles.

Note: There are no AAAs (corners) and SSAs (edges) (exceptions: right triangles are HL, which belong to SSA) in congruence determinations, neither of which can uniquely determine the shape of a triangle.

A is the abbreviation of the English angle, and S is the abbreviation of the English edge.

H is the abbreviation of hypotenuse, and L is the abbreviation of right-angled edge.

There are 4 ways to make a normal triangle and 5 ways to make a right triangle.

1) Corner edges: 2 sides and their angles correspond to equal, and these 2 triangles are congruent. Abbreviated as (2) Corner Corner:

The 2 angles and their edges correspond to the equal, and the 2 triangles are congruent. Abbreviated as (3) corner edges: 2 corners and one of their corners are equal to each other, and these 2 triangles are congruent.

Abbreviated as: (4) Edges and edges: The 3 sides correspond to equal, and these 2 triangles are congruent. Abbreviated as: (5) right-angled side hypotenuse: the hypotenuse and one of the right-angled sides correspond to the same respectively, and these two triangles are congruent. Abbreviated as: (

The first 4 are available for all triangles, and the 5th is only for right triangles.

In general, in the exam, the line segments and angles are equal, and the congruence needs to be proven.

So we can take the reverse mindset.

What conditions are needed to prove perfection?

To prove that such and such a side is equal to such and such an edge, then it is first necessary to prove the congruence of the triangle containing those two sides.

The resulting equation is then applied (AAS ASA SAS SSS HL) to prove the congruence of the triangle.

Sometimes you also need to draw auxiliary lines to help solve problems.

After the analysis is completed, it is necessary to pay attention to the writing format, in the congruent triangle, if the format is not written well, then it is easy to miss the phenomenon.

To verify two congruent triangles, there are generally five methods: edge-edge-edge-edge-sss, corner-edged (SAS), corner-edged (ASA), corner-edged edge (AAS), and right-angled edge (HL) of right-angled triangle.

Judgment method: 1. SSS (side-side-side) (edge-side-side): The triangle corresponding to the three sides is a congruent triangle.

2. SAS (side-angle-side): The triangle corresponding to the two sides and their angles is congruent triangle.

3. ASA (Angle-Side-Angle): Two corners and their edges correspond to equal triangle congruence.

4. AAS (Angle-Angle-Side): The opposite side of two corners and one of their corners corresponds to the congruence of equal triangles.

5. RHS (Right Angle-Hypotenuse-Side) (also known as HL theorem (hypotenuse, right-angled edge)): In a pair of right-angled triangles, the hypotenuse and another right-angled side are equal. (Its proof is based on the SSS principle).

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.

You pigs, you can't be simple, you have it in math books.

Corner Corner Corner Corner Corner Corner Edge Edge.

(1) Corners: abbreviated as (

2) Corners: Abbreviated as (

3) Corner edges: abbreviated as: (

4) Side by side: Abbreviated as: (

5) Right Angled Hypotenuse: Abbreviated as: (Only used for right triangles.)

Common: SSS, ASA, AAS, SAS, special: RT triangle: HL.

SSS, the three sides correspond to equal.

SAS, the two sides and the angles correspond to equal.

As for aas, two corners and one side correspond equally.

ASA, the two corners and the sandwich edges correspond equally.

hl, the straight and hypotenuse sides of the right triangle correspond equally.

In addition, the isosceles triangle can also be obtained by correspondingly equal to one side and one corner, and the principle is that the bottom angle of the isosceles triangle is equal to the waist.

1. Three groups of two triangles with equal sides (SSS for short). Room rent.

2. There are two congruence triangles of hunger chakra (SAS) corresponding to the two sides and their angles.

3. There are two triangles with two angles and their edges corresponding to two triangles of congruence (ASA).

Four ways to prove triangle congruence:

Corner edges, edge edges, corner edges, and corner edges correspond to two equal triangle congruences.

Right triangle with an oblique edge, right edge (HL).

As an auxiliary line: connect BF and extend AF to cross BC to G. It can be proved that BF=CF, AG is the line segment on the middle perpendicular line of the bottom edge of the right-angled isosceles triangle BC.

Angular FBC = Angular FCB. The triangle def is all equal to the triangle dbf, and the angle efd = the angle bfd. Angle EFB = Angle FBC + Angle FCB; Angular EFB = Angular EFD + Angular BFD.

Angle EFD = Angle. ......The triangular AFD is similar to the triangular AGB. The relationship between quantity and location comes out.

With this idea in mind, let's do the rest yourself.