How are rigid transformations used to justify the sas congruence theorem?

The SAS (Side-Angle-Side) Congruence Theorem is a fundamental concept in geometry that states that if two triangles have two sides and the included angle of one triangle congruent to the corresponding parts of another triangle, then the triangles are congruent. In other words, if two triangles share the same side lengths and the same included angle, then they are identical in shape and size.

One way to justify the SAS Congruence Theorem is by using rigid transformations. Rigid transformations are geometric transformations that preserve the size and shape of an object. Examples of rigid transformations include translations, rotations, and reflections.

To use rigid transformations to justify the SAS Congruence Theorem, we start with two triangles that have two sides and the included angle of one triangle congruent to the corresponding parts of another triangle. We then use rigid transformations to superimpose one triangle onto the other, so that they occupy the same space in the plane.

To justify the SAS Congruence Theorem, we can use rigid transformations to superimpose one triangle onto the other. Let’s call the two triangles Triangle ABC and Triangle A’B’C’, where AB = A’B’, AC = A’C’, and angle A = angle A’.

First, we can use a translation to move Triangle A’B’C’ so that side A’B’ is aligned with side AB. This transformation will not change the size or shape of Triangle A’B’C’, but will move it to a different location in the plane.

Next, we can use a rotation to rotate Triangle A’B’C’ around side AB so that angle A’ is aligned with angle A. This transformation will not change the size or shape of Triangle A’B’C’, but will rotate it around the common side.

Finally, we can use another translation to move Triangle A’B’C’ back to its original position, now superimposed on Triangle ABC. Since side AB = A’B’, side AC = A’C’, and angle A = angle A’, the two triangles are congruent by the SAS Congruence Theorem.

The following diagram illustrates the process

  C                  C'
   / \                / \
  /   \              /   \
 /     \            /     \
A-------B         A'------B'
  Triangle         Triangle
     ABC              A'B'C'

Note that after the rigid transformations are applied, Triangle A’B’C’ is in the exact same position and orientation as Triangle ABC, indicating that the two triangles are congruent.