Congruences between a square and itself

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In summary, a square is congruent to itself in all aspects, including all four sides and four right angles. This can be represented by listing all the reflexive congruences between the sides and angles. This can also be seen by the fact that a square can be labeled in multiple ways, but only certain orderings will result in the same square.
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Homework Statement



What are the congruences between a square and itself?

Homework Equations



none

The Attempt at a Solution



Let A, B, C, and D be the sides of the square and e, f, g and h the four right angles.

All four sides are congruent to themselves--same with the angles.

Side A is congruent to Side B
Side A is congruent to side C
Side A is congruent to side D
Side B is congruent to side C
Side B is congruent to side D
Side C is congruent to side D

And we can symmetrically list the congruences between the angles.

And the whole square is congruent to itself...

It's a simple problem but there's no answer key in the text and I'm wondering if I'm right to count all the reflexive congruences...
 
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  • #2
Saying "side A is congruent to side B" does not say the square is conguent to itself.

Since you can "label" a square by listing its vertices in order, ABCD, how many different ways of ordering ABCD give the same square? BCDA does but BACD does not. Do you see why?
 

1. What is a congruence between a square and itself?

A congruence between a square and itself refers to the relationship between two identical squares, where one can be transformed into the other through a series of rotations, reflections, and translations.

2. How do you prove that two squares are congruent?

To prove that two squares are congruent, you must show that they have equal side lengths and equal angles. This can be done through various methods, such as using the side-angle-side (SAS) or angle-side-angle (ASA) congruence criteria.

3. Can two squares with different orientations be congruent?

Yes, two squares with different orientations can still be congruent as long as they have equal side lengths and equal angles. The orientation of a square does not affect its congruence with another square.

4. How many congruence transformations are needed to prove that two squares are congruent?

At least two congruence transformations are needed to prove that two squares are congruent. This can be a combination of rotations, reflections, and translations, as long as they result in the two squares being identical.

5. Are all squares congruent to themselves?

Yes, all squares are congruent to themselves because a shape is always congruent to itself by definition. This means that a square can always be transformed into an identical square through a series of congruence transformations.

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