pexamples of axioms for four line geometry

SSS: the three sides of one triangle are respectively equal to the three sides of the other triangle, or The four standard congruence tests for triangles In the diagram to the right, the two footprintsĪre congruent because one can be reflected onto the other. The example above with the cards involved translationsĪnd rotations. Rotations and reflections so that it fits exactly One can be moved by a sequence of translations, ‘Two plane figures are called congruent if Thus congruent figures can be defined in an alternative way that specifies the allowed transformations: If two figures are congruent, such a movement can always be done by a sequence of translations, rotations and reflections − reflect the first figure in any axis if it has the opposite parity to the second, then translate any point of the first figure to the matching point of the second figure, then rotate the first figure until it fits exactly on top of the second. The word ‘congruent’ comes from Latin and means ‘in agreement’ or ‘in harmony’.ĭefinition: Two plane figures are called congruent if one figure can be moved so that it fits exactly on top of the other figure. Two objects like this are called congruent. We can move one card and place it on top of the other one so that the pictures on the two cards coincide exactly, as shown below. If we take the five of spades from each of two identical decks of cards, they look exactly the same. We will take the first three tests as axioms of our geometry, and will be content to demonstrate their reasonableness using ruler-and compasses constructions. Justifying the congruence tests is no easy matter − the logical problems inherent in them were only sorted out at the end of the nineteenth century, by David Hilbert in particular. The resulting geometric proofs, using mostly only congruent triangles, are clear and straightforward in their logic. Argument based on direct appeals to symmetry, however, is notoriously difficult to construct and to evaluate, and the ancient Greek mathematicians, most famously Euclid, introduced argument based on congruence as a replacement. This is particularly true in geometry, where the elementary figures that we study -like squares, rectangles, circles − exhibit obvious reflection and rotation symmetries.

We will develop the congruence tests as a solution to this question.Ī great deal of mathematics depends on finding and exploiting symmetries. In an analogous way, a certain minimum amount of information is needed to draw a particular triangle.

They must know that everything important about that landscape can be calculated from the measurements that they have taken. It is also true that figures involving curves can be congruent, such as circles of the same radius.Ī good way to think about congruence is to ask, ‘How much information do I need to give someone about a figure if they are going to draw it?’ For example, surveyors go to a lot of trouble taking careful measurements of a landscape. We shall develop the four standard tests used to check that two triangles are congruent. Most of our discussion therefore concerns congruent triangles. (Pythagoras’ theorem gives us the answer 2 cm for this length.) This very simple idea of matching lengths, matching angles, and matching areas becomes the means by which we can prove many geometric results.Ī polygon can always be divided up into triangles, so that arguments about the congruence of polygons can almost always be reduced to arguments about congruent triangles.

For example, if we measure or calculate the unmarked side length of the diagram on the left above, then the matching length is the same in the diagram on the right above. Knowing that two figures are congruent is important. On the other hand, the two figures below are exactly the same in all respects apart from their position and orientation − we can pick up one of them and place it so that it fits exactly on top of the other. For example, all the angles of the square and the rectangle below are right angles, and they have the same area, but their side lengths are different. Two geometric figures may resemble each other in some ways, but differ in others.

Pexamples of axioms for four line geometry