Congruent triangles's judgment 1. Judgment of consistency of general triangles
(1) Edge axiom: three edges correspond to the congruence of two equal triangles ("edge" or "SSS").
(2) Axiom of edges and corners: two triangles with equal included angles ("edges and corners" or "SAS") are congruent.
(3) Angle axiom: Two angles and their sides correspond to the congruences of two equal triangles ("Angle" or "ASA") respectively.
(4) Angle edge theorem: There are two angles, and the opposite side of one angle corresponds to two equal triangles ("Angle edge" or "AAS").
2. Determination of congruence of right triangle
Judging the congruence of a general triangle can prove the congruence of a right triangle.
The hypotenuse and the right-angled side correspond to the coincidence of two right-angled triangles ("hypotenuse, right-angled side" or "HL").
Note: The triangles corresponding to two diagonal angles (SSA) and triangle (AAA) are not necessarily congruent.
The angle associated with the triangle is 1, the inner angle of the triangle.
The sum of the internal angles of a triangle is equal to 180.
2, the outer corner of the triangle
The angle formed by one side of a triangle and the extension line of the other side is called the outer angle of the triangle.
The outer angle of a triangle is equal to the sum of two non-adjacent inner angles.
The outer angle of a triangle is greater than any inner angle that is not adjacent to it.
The line segment 1 is related to a triangle, and the sides of the triangle.
A figure composed of three line segments that are not on the same line end to end is called a triangle. The angle formed by two adjacent sides is called the inner angle of a triangle, which is called the angle of a triangle for short.
A triangle with vertices A, B and C is marked as △ABC and pronounced as "triangle ABC".
The sum of two sides of a triangle is greater than the third side.
2. The bisector of the height, midline and angle of a triangle.
3. The stability of triangle
Triangles are stable.
Similar triangles's judgment method is obviously troublesome, because judging whether two triangles are similar by definition needs to consider six elements, namely, whether the three groups of corresponding angles are equal and whether the three groups of corresponding sides are proportional respectively. So we give the following simple methods to judge the similarity of two triangles:
(1) If two sides of a triangle are in direct proportion to two sides of another triangle, and the included angles are equal, then the two triangles are similar;
(2) If three sides of a triangle are proportional to three sides of another triangle, then the two triangles are similar;
(3) If two angles of a triangle are equal to two angles of another triangle, then the two triangles are similar.
Trilateral relationship of triangle: in a triangle, the sum of any two sides is greater than the third side, and the difference between any two sides is less than the third side.
Let the three sides of a triangle be a, b, c, b and c.
rule
a+b & gt； c
a+c & gt； b
b+ c & gt； a
a-b & lt； c & ltdiv = " " & gt
a-c & lt； b & ltdiv = " " & gt
b-c & lt； a & ltdiv = " " & gt
In a right triangle, let A and B be right angles and C be the hypotenuse.
Then the sum of the squares of two right angles is equal to the square of the hypotenuse.
In an equilateral triangle, a = b = CB = C.
In an isosceles triangle, a and b are two waists, so a = b.
When the opposite sides of the internal angles A, B and C of the triangle ABC are A, B and C respectively, C2=a2+b2-2abcosc.
Similar triangles's so-called similar triangles is the same shape, but different sizes. But as long as the shape is the same, no matter how the size changes, it is similar, so it is called similar triangles.
Two triangles with equal triangles and proportional sides are called similar triangles.
Similar triangles's judgment method is as follows: A straight line parallel to one side of a triangle (or extension lines on both sides) intersects with the other two sides, and the triangle formed is similar to the original triangle.
If two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar.
If the ratio of two sets of corresponding sides of two triangles is equal and the corresponding included angles are equal, then the two triangles are similar.
Two triangles are similar if the ratio of their three corresponding sides is equal.
Theorem for judging the similarity of right-angled triangles 1: The hypotenuse is similar to two proportional right-angled triangles corresponding to one right.
Theorem 2: A right triangle is divided into two right triangles by the height on the hypotenuse, which is similar to the original right triangle, and the two right triangles are similar.
The above is the summary of the key knowledge points of junior high school mathematics triangle that I sorted out for you.