Formulate the congruence theorem for convex quadrilaterals

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Similar to triangles, there is also the possibility of formulating a congruence theorem with convex quadrilaterals. This is of course a little more complicated than with the triangles. But with a little delusion it can be done.

Considerations for Convex Quadrilaterals

Before formulating a congruence theorem, you should first be clear about several things:

  • Convex Quadrilaterals are all quadrilaterals where the diagonals intersect within the quadrilateral.
  • If you formulate a congruence theorem, it must be possible to use this theorem to construct the square. Imagine the values ​​you have to give to a partner on the phone so that they can draw exactly the same convex square that you drew.

The idea that he is on the phone helps you understand that everything needs to be explained verbally. You can't show anything. So instead of “this line there” you have to use specific names.

Preparation to find the congruence theorem

  1. Draw any convex square with its diagonals.
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  2. Label it as you normally would with squares. Start with the lower left corner that you will call A. Driving in alphabet by naming the remaining corners counterclockwise.
  3. The route from A to B is a, the route from B to C is b, and so on. The angle at A is alpha, the angle at B beta etc. The distance AC is d1 and the distance BD is d2.
  4. If you now want to formulate a congruence theorem for the convex square, you should put them all together and measure angles, then it will be easier to check whether you have found a congruence theorem.

Derivation of a congruence theorem of convex quadrilaterals

  1. Start with SSSS according to the SSS congruence theorem for triangles. You will quickly find that you cannot draw a specific convex square with these sizes. If you don't know an angle, you won't be able to draw the auxiliary triangle ABC or BCD. Consider that a square can have the same side length as a diamond, so you cannot set up a congruence theorem for quadrilaterals with only sides.
  2. Try it with 3 sides and 2 angles, SWSWS, for example a, beta, b, gamma and c. You will quickly see that you can construct the triangle ABC from a, beta and b (congruence theorem SWS). Now you can draw the angle gamma on the segment b at point C and plot the length c on the free leg of gamma. You get point D. So your partner on the phone can draw the square.
  3. So there is a connection between the congruence sets of triangles and squares. Think about how the auxiliary triangle ABC can still be constructed. You could also do it through d1, a, b (SSS) or WSW. In both cases you would need to know lines or angles that have nothing to do with the 4 sides and 4 angles of quadrilaterals. In this context, the auxiliary triangle is only to be constructed according to SWS.
  4. Now consider what other possibilities there are to construct quadrilaterals from the triangle ABC. Instead of gamma, you could also know the angle alpha and the distance d. You would then have d, alpha, b, beta, c so again SWSWS. In general, the theorem of congruence then reads: three sides and two angles in between.
  5. You can of course also - based on the auxiliary triangle ABC - know the angle gamma and the distance d. In this case you have to plot the angle gamma on segment b and draw a circle around A with radius d. You will get an intersection at D. So SSWSW is also a congruence theorem for convex quadrilaterals.

If you do the deliberations with the auxiliary triangle BCD or assume that you have alpha, a, beta, b and c, that also comes back to SSWSW, which you also call 3 pages and one of the pages attached 2 angle can denote.

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