Similar triangles's judgment method
Two similar triangles should write letters representing the corresponding vertices at corresponding positions. If "△ABC is similar to △DEF" in written language, it means that the corresponding vertices of these two triangles are not written in corresponding positions, while if "△ABC∽△DEF" in symbolic language, it means that the corresponding vertices of these two triangles are written in corresponding positions.
Method 1 (preparatory theorem)
The straight line parallel to one side of the triangle intersects with the other two sides (or the extension lines of both sides), forming a triangle similar to the original triangle; (This is similar triangles's judgment lemma, which is the basis of the following judgment methods. The proof method of this lemma needs the proof that parallel lines are proportional to line segments)
Method 2
If two angles of one triangle are equal to two angles of another triangle,
Then these two triangles are similar.
Method 3
Two triangles are similar. If the ratio of their corresponding two sides is equal, the corresponding included angles are also equal.
Method 4
Two triangles are similar if the ratio of their three sides is equal.
Method 5 (Definition)
Two triangles with equal corresponding angles and proportional corresponding sides are called similar triangles.
Edit a triangle that must be similar.
1. Two congruent triangles must be (absolutely) similar.
2. Two isosceles right triangles must be (absolutely) similar.
(Two isosceles triangles are similar if the top or bottom angles are equal. )
3. Two equilateral triangles must be (absolutely) similar.
Edit this right triangle similarity judgment theorem.
1. The hypotenuse is similar to two proportional right-angled triangles corresponding to right-angled sides. 2. The 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.
Edit the judgment theorem inference of triangle similarity in this paragraph.
Inference 1: Two isosceles triangles with equal top or bottom angles are similar. Inference 2: waist and bottom are similar to two isosceles triangles in proportion. Inference 3: Two right-angled triangles with equal acute angles are similar. Inference 4: Two right triangles divided by the height on the hypotenuse are similar to the original triangle. Inference 5: If the median lines of two sides and one side of a triangle are proportional to the corresponding parts of another triangle, then the two triangles are similar. Inference 6: If the median lines of two sides and the third side of a triangle are proportional to the corresponding parts of another triangle, then the two triangles are similar.
Edit the properties of similar triangles in this paragraph.
1. The corresponding angles of similar triangles are equal, and the corresponding edges are proportional. 2. The ratio of all corresponding line segments (corresponding height, corresponding midline, corresponding angle bisector, circumscribed circle radius, inscribed circle radius, etc.). ) is equal to similarity ratio in similar triangles. 3. The ratio of similar triangles perimeter is equal to the similarity ratio. 4. The ratio of similar triangles area is equal to the square of similarity ratio.
Edit this paragraph as a special case of similar triangles-congruent triangles.
The similarity ratio is 1, the corresponding angles are equal, the corresponding sides are equal, the perimeter is equal, and the area ratio is equal.