Such are the Konkruenzsätze.
Considerations convex quadrangles
Before formulating a congruence, you should first be on Different clear:
- Convex quadrilaterals are all rectangles in which intersect the diagonal inside the square.
- If you formulate a congruence, it must be possible, using this theorem to construct the quadrangle. Imagine what values you need to call a partner on the phone so that it can draw exactly the same convex quadrilateral that you have drawn.
The idea that this is on the phone, helps to understand that everything must be explained verbally. You can not show anything. So sick of "this line because" you have to use some concrete names.
find preparation to the congruence
- Characters, any convex quadrilateral and its diagonals.
- Label it as is common in the four corners. Start with the bottom left corner, you call A. Driving in the alphabet continues by calling counterclockwise the remaining corners.
- The route that leads from A to B, is a, which is from B to C b and so on. The angle A is in Alpha, Beta, etc. The B in the segment AC is d 1, and the distance BD is d 2.
- If you now want to formulate a congruence of the convex quadrilateral, you should all stick and angle measure, then it is easier to check if you have a congruence found.
Deriving a Kongruenzsatzses of convex quadrangles
- Start with SSSS according to the congruence SSS in triangles. You will soon discover that you can not draw a definite convex quadrilateral with these sizes. If you do not know any angle, you are the auxiliary triangle ABC or BCD can not draw. Consider a square may have the same side length as a diamond, so only pages you can place no congruence of squares.
- Try it with 3 sides and 2 angles, SWSWS, so for example, a, beta, b, c and gamma. You quickly realize that you can construct from a, b and beta, the triangle ABC (congruence SWS). Now you can at the distance b at the point C draw the angle gamma and the free limb of the length Gamma c erode. You-find point D. Your partner on the phone so it can draw the rectangle.
- So there is a relationship between the congruence of triangles and quadrangles. Consider how the auxiliary triangle ABC can be constructed. You could also by d 1, a, b (SSS) or WSW determine. In both cases, you would need to know distances or angles that have nothing to do with the 4 sides and 4 angles of quadrangles. The auxiliary triangle is thus to construct in the context only by SWS.
- Now consider what other options there are to construct squares of the triangle ABC. You could instead Gamma also know the angle alpha and the distance d. Then they would have d, alpha, b, beta, c So again SWSWS. General is the congruence then: three sides and 2 intermediate angle.
- Of course you can - based on the auxiliary triangle ABC - know the angle gamma and the distance d in the case you need to b antragen at the track the angle gamma and draw a circle around A with radius d.. You get a point of intersection with D. So also SSWSW is a congruence of convex quadrangles.
If you make the considerations to the auxiliary triangle BCD or assume that Alpha, a, beta, b and c have also runs the back out on SSWSW what to designate as 3 sides and one of the sides adjacent 2 angle.