How to determine a similar triangle and how to determine a similar triangle?

The determination theorem of similar triangles has the following conclusions:

Theorem The two angles correspond to two equal triangles that are similar.

Theorem Two triangles with proportional sides and equal angles are similar.

Theorem Two triangles proportional to three sides are similar.

Theorem Two right-angled triangles with a right-angled side proportional to the hypotenuse are similar.

Based on the above decision theorem, the following conclusions can be derived:

Corollary: The three sides correspond to two triangles that are parallel.

Corollary The two sides of a triangle and the midline on either side of the triangle are proportional to the corresponding part of the other triangle, then the two triangles are similar.

A special case of similar triangles.

1.All congruent triangles are similar.

Congruent triangles are special similar triangles with a similarity ratio of 1. Conversely, when the similarity ratio is 1, the similarity triangle is congruent.

2.There is a two isosceles triangles with equal top or bottom angles, both of which are similar.

As a result, all equilateral triangles are similar.

1) The straight line parallel to one side of the triangle and the triangle formed by the other two sides are similar to the original triangle;

2) If the ratio of the corresponding sides of the two triangles is equal and the angles are equal, the two triangles can also be similar (abbreviated as: the two sides are proportional and the angles are equal, and the two triangles are similar. )；

3) If the three sides of a triangle correspond to the three sides of another triangle, then the two triangles are similar (abbreviated: the three sides correspond to proportion, and the two triangles are similar. )；

4) If the two angles of two triangles correspond to each other (or the three corners correspond to each equal), then two triangles are similar (abbreviated as two angles corresponding to equal, and two triangles are similar).

The method of determining similar triangles has its decision theorem. For two triangles, the two angles correspond to the same and the two triangles are similar. The three sides correspond to two triangles that are proportional.

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1) A straight line parallel to one side of the triangle intersects with the other two sides and the extension lines on both sides, and the triangle formed is similar to the original triangle.

2) If the two sides of one triangle and the two sides of another triangle correspond proportionally, and the angles are equal, then the two triangles are similar.

3) If the three sides of a triangle correspond to the three sides of another triangle, then the two triangles are similar.

4) If the two corners of two triangles correspond to each equal (or the three angles correspond to each equal), then there are two triangles that are similar.

Triangle similarity:

1. The two angles correspond equally, and the two triangles are similar.

2. The two sides correspond proportionally and the angles are equal, and the two triangles are similar.

3. The three sides correspond to proportions, and the two triangles are similar.

4. If the hypotenuse and one right-angled side of a right-angled triangle correspond to the hypotenuse and a right-angled side of another right-angled triangle.

A straight line parallel to one side of the triangle cuts off the other two sides of the line, and the resulting triangle is similar to the original triangle. (This is the theorem of the similarity triangle determination, which is the basis for the following determination method proof.) The proof method of this lemma requires the proof that the parallel lines are proportional to the line segments. ）

Corner relationship of a triangle:

1. Sine theorem: a sina = b sinb = c sinc sinc 2, cosine theorem:

a²=b²+c²-2bccosa

b²=a²+c²-2accosa

c²=a²+b²-2abcosa

3. Tangent theorem:

tan[(a-b) 2]= tan(c 2) (a-b) (a+b) or (a+b) tan[(a-b) 2]=(a-b)tan(c 2) or (a+b) tan[(a-b) 2]=(a-b) tan[(a+b) 2].

Triangle Judgment:If the three sides of a triangle correspond proportionally to the three sides of another triangle, then the two triangles are similar; If the two sides of a triangle correspond to the two sides of another triangle and the angles are equal, then the two triangles are similar.

If the two corners of a triangle correspond equally to the two angles of another triangle, then the two triangles are similar. If the hypotenuse and one right-angled edge of a right-angled triangle are proportional to the hypotenuse and one right-angled side of another right-angled triangle, then the two triangles are similar.

A similar triangle is two triangles with three equal angles and proportional sides. The decision theorem is as follows:

1. The two corners correspond to two equal triangles similar to the hall god.

2. The two triangles that are proportional on both sides and have equal angles are similar.

3. Two triangles with proportional sides are similar.

4. Two right-angled triangles with a right-angled side proportional to the hypotenuse are similar.

Similar triangle It mainly describes the relationship between sides and corners in a similar triangle. It is one of the important proof models in geometry, and it is the generalization of congruent triangles. Congruent triangles can be understood as similar triangles with a similarity ratio of 1.