One, two graphics with the same shape must be hand in hand.

Conclusion: If the shapes of two figures are the same, but they are not necessarily completely coincident, then you can definitely hold hands through translation.

Proof process:

1. First, suppose there are two graphs G and H, which have the same shape but do not hold hands.

2. Secondly, the graph G moves to another position along a certain direction, and its relative position with the original graph H has changed.

3. Because the two figures have the same shape, but their positions have changed, they should still be overlapped by translation. This contradicts the hypothesis.

Therefore, two figures with the same shape must be hand in hand.

Second, the translation distance of hand-in-hand model is equal.

Conclusion: If two graphs G and H are hand-in-hand models, the translation distance between corresponding points is equal.

Proof process:

1. Suppose two graphs G and H are hand-in-hand models.

2. Let the distance from any point P on the graph G to the corresponding point Q be d 1, and the distance from the corresponding point R to the corresponding point S on the graph H be d2.

3. Because the two figures are hand-in-hand models, the distance between the corresponding points is equal, that is, d 1=d2.

4. Therefore, the translation distance is equal.

Third, the angle change law of hand-in-hand model

Conclusion: If two graphs G and H are hand-in-hand models, the angles formed by the corresponding points are equal.

Proof process:

1. First, select any two corresponding points A and B on the graph G, and select corresponding points C and D on the graph H. ..

2. Connect AB and CD to form a diagonal line.

3. Because the two figures are hand-in-hand models, the distance between AB and CD is equal, that is, the diagonal length is equal.

4. So according to the properties of equidistant diagonal lines, the angles formed by the corresponding points are equal.

Fourthly, hand-in-hand mode and symmetry.

Conclusion: If a graph takes a certain point as the center of symmetry, then its hand-in-hand model also takes that point as the center of symmetry.

Proof process:

1. Suppose there is a graph G with a point o as the center of symmetry.

2. Choose any pair of corresponding points A and B to connect OA and OB.

3. Because the graph G is centered on the O point, A and B are symmetrical about the O point.

So it can be seen from the first conclusion that the two figures where A and B are located must be hand in hand. Therefore, its hand-holding mode is also around this point.