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05.02.02 - Congruence and Rigid Motions (Geometry)

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The main point of this lesson is that if two figures coincide, then they are congruent. Rigid motions transform the figures without changing the sides or the angles. Therefore, if you can apply rigid motions to transform two figures so that they coincide, these figures are congruent.

Cool huh?

Okay, so let's try this.

Consider the figures shown here.




The red heptagon looks like the magenta heptagon but are they really congruent?

Start by rotating the red heptagon about its 'center.' The images below show the red heptagon rotated by approximately sevenths of a circle so that the base is horizontal.




The second figure, approximately two sevenths of a rotation, looks promising. None of the others look correct.

Now back to the original image. If we rotate the red heptagon approximately two sevenths of a circle, we obtain the following.




If we translate the red figure to the left and down a bit, we get the following.




So, yes. By using rigid motions and the coinciding axiom, we have proven that the original red heptagon is congruent with the magenta heptagon.

Try another?




These images also look like they may be congruent. Try rotating the blue image so that the bases are parallel.




That does look like the images may align. Translate the blue figure to the left.




Okay, so that doesn't quite work. Should we give up? Of course not! Why have another rigid transformation to try. Reflect the blue image across the line shown here.




And tuh duh! We did it! The blue and green figures are actually congruent!

Okay, consider the following counterexample.




Try rotating the magenta parallelogram about the point shown below.




Okay, not quite right. What if we reflect it across the pair of lines shown below.




Did you get the exact same image? Me, too.

The only other rigid motion we have is translation and it should be obvious that this motion will not improve the situation.

Therefore, we are done. We have attempted all reasonable permutations of the three rigid motions. There is nothing else to try. These images are not congruent.