How study of Congruence be helpful for us
Answers
Step-by-step explanation:
Two geometric figures may resemble each other in some ways, but differ in others. 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.
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. Such figures are called congruent.
Knowing that two figures are congruent is important. 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. (Pythagoras’ theorem gives us the answer 2cm 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.
A 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. Most of our discussion therefore concerns congruent triangles. We shall develop the four standard tests used to check that two triangles are congruent. It is also true that figures involving curves can be congruent, such as circles of the same radius.
A 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. They must know that everything important about that landscape can be calculated from the measurements that they have taken. In an analogous way, a certain minimum amount of information is needed to draw a particular triangle. We will develop the congruence tests as a solution to this question.
A great deal of mathematics depends on finding and exploiting symmetries. This is particularly true in geometry, where the elementary figures that we study —like squares, rectangles, circles − exhibit obvious reflection and rotation symmetries. 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. The resulting geometric proofs, using mostly only congruent triangles, are clear and straightforward in their logic.
Step-by-step explanation:
For two polygons to be congruent, they must have exactly the same size and shape. This means that their interior angles and sides must all be congruent. ... That's why studying the congruence of triangles is so important--it allows us to draw conclusions about the congruence of polygons, too.