definition
Two triangles that can completely coincide are called congruent triangles. (Note: congruent triangles is a special case of similar triangles's similarity ratio 1: 1) When two triangles completely coincide, the overlapping vertices are called corresponding vertices, the overlapping edges are called corresponding edges, and the overlapping angles are called corresponding angles. It can be concluded that the corresponding edges of congruent triangles are equal and the corresponding angles are equal. (1) The opposite side of the corresponding corner of congruent triangles is the corresponding edge, and the edge sandwiched by two corresponding corners is the corresponding edge; (2) The diagonal of the corresponding side of congruent triangles is the corresponding angle, and the included angle of two corresponding sides is the corresponding angle; (3) If there is a male party, the male party must be the corresponding party; (4) If there is a common angle, it must be the corresponding angle; (5) If there is an antipodal angle, the antipodal angle must be the corresponding angle;
Three groups of two triangles with equal sides are congruent (SSS or simply "edge"), which also explains the stability of triangles.
There are two congruent triangles (SAS or "corner sides"), and the two sides and their included angles correspond to each other.
3. Two triangles with two corners are congruent with their clamping edges (ASA or "corners").
Push from 3 o'clock
4. Two triangles with two angles and opposite sides of one angle are congruent (AAS or "corner edge").
5. The congruence conditions of right-angled triangles are as follows: the hypotenuse and right-angled side correspond to the congruence of two right-angled triangles (HL or "hypotenuse and right-angled side"). So SSS, SAS, ASA, AAS, HL are all theorems for judging triangle congruence. Note: In congruence judgment, there is no AAA angle and SSA (special case: right triangle is HL, belonging to SSA) edges and corners, and the shape of triangle cannot be uniquely determined in both cases. A is the abbreviation of English corner, and S is the abbreviation of English corner. H is the abbreviation of hypotenuse, and l is the abbreviation of leg. 6. Three median lines (or bisectors of height and angle) respectively correspond to the congruence of two equal triangles.
Edit the properties of this paragraph.
The condition of triangle congruence: 1, and the angles corresponding to congruent triangles are equal. 2. The corresponding sides of congruent triangles are equal; 3. The corresponding vertices of congruent triangles are equal. 4. The heights of the corresponding sides of congruent triangles are equal. 5. The bisectors of the corresponding angles in congruent triangles are equal. 6. The median lines corresponding to congruent triangles are equal. 7. congruent triangles have equal areas. 8. The circumference of congruent triangles is equal. 9. congruent triangles can completely overlap. The method of triangle congruence: 1, two triangles with equal sides are congruent. (SSS) 2。 Two sides and their included angles correspond to the congruence of two triangles. (SAS) 3。 Two corners and their clamping edges correspond to the coincidence of two triangles. (ASA) 4。 Two equilibria (AAS) 5 of a triangle with two angles and their opposite sides corresponding to one angle. The two equilibria of a right-angled triangle correspond to a hypotenuse and a right-angled side. (HL)
Edit this inference
In order to verify the congruent triangles, it is not necessary to verify that all edges and all angles are correspondingly the same. The following judgment consists of three corresponding parts, that is, congruent triangles can be judged by the following definition: S.S.S (edge-edge-edge): If the three sides of each triangle are correspondingly equal in length, then two triangles are congruent. S.A.S (Edge-Angle-Edge): If the lengths of two sides of each triangle are correspondingly equal and the angles between the two sides are correspondingly equal, then the two triangles are congruent. A.S.A (Angle-Edge-Angle): If the two angles of each triangle correspond equally and the edges between the two angles correspond equally, then the two triangles are congruent. A.A.S (Angle-Angle-Edge): If the two angles of each triangle correspond equally, and the edges not caught by the two angles correspond equally, then the two triangles are congruent. R.H.S./H.L (right angle-hypotenuse-side): If the right angle, hypotenuse and the other side of each triangle are equivalent, then the two triangles are identical. But we can't use any three equal parts to judge whether a triangle is congruent. The following judgment is also applicable to the bisection of two triangles, but congruent triangles cannot be judged: A.A.A (Angle-Angle-Angle): Any three angles of each triangle correspond equally, but congruent triangles cannot be judged, but similar triangles can be judged. A.S.S (Angle-edge-edge): One angle of each triangle is equal, and the other two sides are equal (excluding the angle between them), but this does not determine the congruent triangles unless it is a right triangle. But if it is a right triangle, it should be judged by R.H.S
Edit the usage of this paragraph.
1, the congruence of triangle is a condition, and the conclusion is that the corresponding angles and sides are equal. But the judgment of congruence is just the opposite. 2. It is the key to learn to find out the corresponding edges and angles in two congruent triangles accurately by using properties and judgments. When writing the congruence of two triangles, the corresponding vertices, angles and edges must be written in the same order, which provides convenience for finding the corresponding edges and angles. 3. When there are more than two equilateral triangles in the graph, we should first consider using SAS to find congruent triangles. 4. In practice, we usually use congruent triangles to measure equidistant distance. And can be used in industry and military affairs. Triangle has certain stability, so we use this principle to make scaffolding and other supports.
Edit this paragraph to do the problem skills
Generally speaking, it is necessary to prove the congruence when the line segment and angle are equal in the exam. Therefore, we can adopt the way of reverse thinking. In order to prove congruence, what conditions are needed to prove that so-and-so side is equal to so-and-so side? First, we must prove that the triangle containing those two sides is congruence. Then the triangle congruence is proved by the obtained equation (AAS/ASA/SAS/SSS/HL). Sometimes you need to draw auxiliary lines to help solve problems. Pay attention to the writing format after the analysis. In congruent triangles, if the format is not well written, it is easy to miss it. For example 1, as shown in the figure, it is known that CD⊥AB is in D, BE⊥AC is in E, △ Abe △ ACD, ∠ C = 20, AB= 10, AD= 4, and G is a point on the extension line of AB. Find ∞ △ABE △ ACD, the external angle of △ABE ∠EBG or the adjacent complementary angle of △ Abe ∠ EBG. (2) By using the equivalent property of congruent triangles correspondence angle and the knowledge of external angle or adjacent complementary angle, it is found that ∠EBG is equal to 160. (3) By using the property that congruent triangles's corresponding edges are equal and the relation of equivalent minus equivalent difference, we can get: Ce = Ca-AE = Ba-AD = 6. Solution: ∫△ Abe △ ACD ∠ C = 20 (known) ∴∠ Abe = ∠ C = 20. ∫△Abe?△ACD (known) ∴AC=AB (congruent triangles corresponding sides are equal) AE=AD (congruent triangles corresponding sides are equal) ∴ CE = Ca-AE = Ba-AD = 6 (equality property).
Example analysis of editing this paragraph
Example 1: (Zhejiang Jinhua, 2006) As shown in figure 1, in △ABC and △ABD, AD and BC intersect at point O, ∠ 1=∠2, please add a condition (no more line segments, no more labels or other letters) to make. Explain △ ABC △ bad according to the graphic idea first. It is already known in the title that ∠ 1 = ∠ 2 and AB = AB, only a set of opposite sides are equal or a set of diagonal lines are equal. Solution: the condition of addition is: BC=AD. Proof: in △ABC and △BAD, ∠ 67. ∠a =∠a '∴△ABC?△bad(SAS)。 ∴ AC=BD。 Summary: This question examines congruent triangles's judgment and nature, and the answer is not unique. If the condition is added in one of the following ways: ①BC=AD, ②∠C =∞. Therefore, there is AC=BD. Second, comprehensive open example 2 (Panzhihua, 2006) is shown in Figure 2, with point E on AB and AC=AD. Please add a condition to make congruent triangles exist in the diagram. And give a proof. The condition of adding is _ _ _ _ _ _ _ _. You get a pair of congruent triangles: △△ Proof: Analysis: Under known conditions, a group of edges are equal and there is a common edge in the graph. Therefore, if the included angles of these two sides are equal or the other set of opposite sides are equal, the congruent triangles can be obtained. Solution: the added condition is CE=ED. The congruent triangles obtained is △ CAE △ DAE. It is proved that in △CAE and △DAE, AC=AD, AE=AE and CE=DE, so △ CAE △ DAE (SSS).