Trick of adding guides when proving triangle congruence

1. When proving the congruence of the triangle, if the midline of the triangle is encountered, the midline can be doubled to make the extension line equal to the length of the original midline, and the congruent triangle of the base reed is constructed;

2. Truncate and make up for the shortcomings to make them equal to specific line segments, and then use the relevant knowledge of congruent triangles to solve problems;

3. Using the property of the angle bisector, you can make perpendicular lines from a certain point on the angle bisector to both sides of the angle, and then use the property theorem or inverse theorem of the angle bisector;

4. See the midpoint with the median line, and skillfully use the nature of the median line;

5. Make a specific parallel line at a certain point on the figure to construct a congruent triangle;

6. With the help of the "three-in-one" property of the isosceles triangle, the congruent triangle is constructed;

7. When there is a height, the figure is folded in half with the height as the axis of symmetry to construct a congruent triangle;

8. Complete the figure, find the equivalence relationship of the wheel, and construct the congruent triangle.

Congruent triangles are a very important chapter in junior high school geometry, and many children do not know how to add auxiliary lines to geometric shapes. Below I have sorted out the common ways to add congruent triangle auxiliary lines for your reference.

1. Double-length midline (or quasi-midline) method

In geometry problems, if you encounter the midline of a triangle, the midline of the class, or the line segment related to the midpoint, the method of doubling the length of the midline or the double-length centerline of the class is usually considered to construct a congruent triangle.

2. Truncation method

If the relationship between the sum, difference, multiple, and division of the proved line segment is encountered, the truncation method is usually considered to construct a congruent triangle. Truncation is the process of taking one of the longer segments equal to one of the other two, and then proving that the remaining part is equal to the other. Supplementation is to extend a longer segment, and the extended part is equal to another segment, and then prove that the new segment is equal to the longer segment; Or extend a longer segment equals a longer segment, and then prove that the extension is equal to another segment.

3. The bisector of the encounter angle is used as a double perpendicular line method

When you meet the bisector of angles in the question, make a double perpendicular, and you will have a congruent triangle. You can make perpendicular lines from points on the bisector of angles to both sides, or you can cross points on bisectors to make perpendicular lines of angular bisectors that intersect both sides of the angle.

4. Make parallel line method

In the proof of geometric problems, the method of making parallel lines is also very practical, generally speaking, in the special three solutions such as isosceles and equilateral, making parallel lines is definitely the primary consideration.

People say that geometry is very difficult, and the difficulty lies in the auxiliary line, auxiliary line, how to add it? Grasp the theorem and concept, but also study hard, find out the law by experience, there are angular bisector lines in the diagram, you can lead the perpendicular line to both sides, you can also fold the diagram in half to see, symmetry after the relationship is present, the angle bisector parallel line, isosceles triangle to add, angle bisector plus perpendicular line, three lines in one try to see, the line segment is perpendicular bisectoral, often to both sides of the line connection, to prove that the line segment is double and half, extended and shortened can be tested, the triangle in the two midpoints, the connection is the median line, there is a middle line in the triangle, extend the middle line and other median lines.

1. The method of doubling the length of the midline is to double the midline of the triangle in order to construct a congruent triangle, so as to use the relevant knowledge of congruent triangles to solve the problem. The process of the double-length midline method: extend so-and-so to a certain point, so that so-and-so is equal to so-and-so, so that what is equal to what (the extended one), use SAS to prove the most important point of the congruence (to the vertex angle) double-length midline, extend the midline by double, and complete the construction of the SAS congruent triangle model.

2. Truncate and make up for the shortcomings to make them equal to specific line segments, and then use the relevant knowledge of congruent triangles to solve problems;

Truncation method: (1) cross a certain point to make a perpendicular line on the long side. (2) Intercept a line segment on the long side that is the same as a short side, and then prove that the remaining line segment is equal to the other short side.

Short-fitting method: (1) Extend the short side. (2) Twist the two short sides together by rotating, etc.

3. Using the property of the angle bisector, you can make perpendicular lines from a certain point on the angle bisector to both sides of the angle, and then use the property theorem or inverse theorem of the angle bisector;

4. See the midpoint with the median line, and skillfully use the nature of the median line;

5. Make a specific parallel line at a certain point on the figure to construct a congruent triangle;

6. With the help of the "three-in-one" property of the isosceles triangle, the congruent triangle is constructed;

7. When there is a height, the figure is folded in half with the height as the axis of symmetry to construct a congruent triangle;

8. Complete the figure, find the equiquantity relationship, and construct the congruent triangle.

The first is called the doubling midline congruence. What does it mean, that is, when the geometric feature of the midline appears in the known conditions of the problem, in the case that we cannot find a good breakthrough in the initial image, we can consider extending the midline (generally doubling the extension to form equal sides) to construct a congruent triangle, so as to find out more available conditions and find another way to solve the problem.

The second is called the truncation method. As the name suggests, it is to intercept or extend a section on a certain line segment or edge, so that it constitutes a special feature (generally equal), so that some corner relationships of congruent triangles can be constructed, which is especially suitable for proving the sum, difference, multiple, and division of line segments. For example, in the following question, verify the edge length and relationship of BE+CF>2AD.

The third is to use the three-in-one nature of an isosceles triangle to construct a congruent triangle. We know that the high line on the bottom edge of an equilateral triangle is also the middle line and the angular bisector (three lines in one), so when the isosceles triangle appears in the problem or you can find the isosceles triangle through simple geometric relations, you can try to make this special line to help you think, for example, take the midpoint e e of ab in the next question, and connect de to get some judgments and properties of this special line segment and congruent triangle.

Fourth, using the nature of the angular bisector, we know that if we make perpendicular lines on either side of the angular bisector at any point, the two line segments are of equal length, and if we construct it this way, it is equivalent to obtaining some special angular relations as our thinking group.

Fifth, the "translation" or "flip folding" in the congruent transformation is constructed by using the bisector property of the angle, so that some congruent triangles can also be easily formed, so as to obtain some key hidden conditions to solve the problem. This is a kind of auxiliary line idea that is difficult to think of, and you can understand it in detail through the following question.

The congruent triangle guide practices are summarized as follows:

1.When encountering an isosceles triangle, it can be used as the height on the bottom edge, and the property of "three lines in one" is used to solve the problem, and the thinking mode is to construct a congruent triangle by the "folding" method in the congruent transformation.

2.When encountering the middle line of the triangle, the double length of the middle line, so that the extension line segment is equal to the length of the original middle line, and the congruent triangle is constructed.

3.In three ways to add auxiliary lines to the angle bisector, (1) you can make perpendicular lines from a certain point on the angle bisector to both sides of the angle, and the thinking mode used is the "folding" in the congruent transformation of the triangle, and the knowledge points examined are often the property theorem or the inverse grip theorem of the angle bisector (2) you can make the perpendicular line of the angle bisector at a point on the angle bisector and intersect both sides of the angle to form a pair of congruent triangles. (3) Two points can be cut on both sides of the angle, and two points can be cut at the position of equal length from the vertices of the angle, and then from these two points to a certain point on the bisector of the angle as an edge to construct a pair of congruent triangles.

4.The idea of contouring a congruent triangle is constructed by making a specific bisector at a certain point on the graph, using the mode of thinking of "panning" or "flipping and folding" in congruent transformations.

5.The truncation method and the shortness method, the specific method is to intercept a line segment on a certain line segment to be equal to a specific line segment, or to extend a certain line segment, which is equal to a specific line segment, and then use the relevant properties of the triangle congruence to explain This method is suitable for proving the sum of line segments, the difference, the multiple, the classification of the class.

6.If you know the perpendicular bisector of a line segment, you can connect the two ends of the line segment at a certain point on the perpendicular bisector to create a pair of congruent triangles.