Congruence Through Superposition

 

ABOUT:

If two triangles have two sides equal to two sides, respectively, and have the angle(s) enclosed by the equal straight lines equal, then they will also have the base equal to the triangle, and the remaining angles subtended by the equal sides will be equal to the corresponding remaining angles.

To Prove this

Let ABC and DEF be two triangles having the two sides AB and AC equal to the two sides DE and DF, respectively.

i.e., AB to DE, and AC to DF.

  • Angle BAC be equal to the angle EDF.
  • BC is also equal to the base EF,
  • Triangle ABC is equal to triangle DEF
  • Angle ABC is equal to DEF
  • Angle ACB is equal to DFE.

If triangle ABC is applied to triangle DEF, the point A being placed on the point D, and the straight-line AB on DE, then the point B will also coincide with E, on account of AB being equal to DE.

Since AB coinciding with DE, the straight-line AC will also coincide with DF, on account of the angle BAC being equal to EDF.

The point C will also coincide with the point F, again on account of AC being equal to DF but, point B certainly also coincided with Point E, so that the base BC will coincide with the base EF.

If B coincides with E, and C with F, and the base BC does not coincide with EF, then two straight-lines will encompass an area which is impossible. Thus the base BC will coincide with EF, and will be equal to it. So the whole triangle ABC will coincide with whole triangle DEF, and will be equal to it . And the remaining angles will coincide with the remaining angles, and will be equal to them . And the remaining angles will coincide with the remaining angles, and will be equal to them.

i.e.,ABC to DEF, and ACB to DFE

The method of proof used in this proposition is sometimes called "superposition."

By using the method of superposition we can say that two figures are congruent if and only if they can be superimposed upon each other through a series of rigid transformations so that they are identical i.e. they coincide at in every point.

Application:

Consider real life objects like bottle caps, scales, stamps, etc , when super imposed and every point coincide then they are also congruent, then laws of congruency can be used on the objects to measure.