In junior high school mathematics, auxiliary line is a commonly used problem-solving method, which can help us understand the problem more clearly and find the breakthrough point. The following are some common situations and detailed instructions on how to add auxiliary lines.

Parallel lines are often used to construct parallelograms.

When we meet parallel lines in a graph, we can often construct a parallelogram by making parallel lines. This method is usually used to prove geometric problems such as parallelism or equal angle. For example, if we have a parallelogram ABCD and want to prove AD//BC, we can make a parallel line EF, so we can prove EF//AD by the properties of parallel lines.

There is a line segment with the midpoint of one side of the parallelogram as the endpoint, which is often extended.

When a line segment of the midpoint of one side of a parallelogram appears in the topic, we can usually construct a congruent triangles or isosceles triangle by extending this line segment. For example, in the parallelogram ABCD, point E is the midpoint of BC, and we can extend DE to point F so that EF=BE, so that we can construct an congruent triangle BFE and a triangle DCE, thus solving related problems.

A rectangle or parallel line that can be used as a vertical line when vertical.

When there is a vertical line in the topic, we can construct a rectangle or parallel line by making a vertical line. For example, in the triangle ABC, the angle ACB=90 degrees, we can make a vertical line CD, so that CD is perpendicular to AB, so that we can construct a rectangular ACBD, and we can solve related problems by using the properties of the rectangle.

The distance between a point on one diagonal of a square and two ends on the other diagonal is equal.

In a square, the distance between a point on one diagonal and two ends on the other diagonal is equal. This property is often used to prove that some points are the midpoint of a graph. For example, in a square ABCD, point E is the midpoint of AD, and we can use this property to prove that point E is also the midpoint of BD.

The midpoint of one side of a square is often taken as the midpoint of the other side.

When the midpoint of a square appears in the topic, we can usually take the midpoint of the other side to construct a congruent triangles or isosceles triangle. For example, in a square ABCD, point E is the midpoint of BC, and we can take the midpoint F of AD, so that we can construct an congruent triangle BEF and a triangle DCE, thus solving related problems.

Rotate and transform with a square.

The rotation of a square is a very useful technique, which can help us prove that some angles are equal or some line segments are equal. For example, we can rotate the square ABCD around its center to make it coincide with the original figure, which can prove that some angles are equal or some line segments are equal.

The above are some common ways to add auxiliary lines, but this is not all. Different topics need different auxiliary lines to help solve them. So we need to be familiar with these methods through a lot of practice and learn how to choose the most suitable method according to the situation of the topic.