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Please consider the corresponding dimensions same as it is a handmade diagram.

My doubt lies at the fact whether superimposability is must for congruence. If it is true, then 2D mirror images can't be congruent since they can't be superimposed without rotating them in 3D space.

Sagar Upadhyay
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    First sentence: https://en.wikipedia.org/wiki/Congruence_(geometry) – Dave Oct 13 '17 at 15:54
  • They are congruent. If they were not rendered congruent, then many triangle congruences we use in proof don't work, making such proofs much more difficult. One example is where we split an isosceles triangle into two mirror image right triangles and use those to prove the base angles congruent. – Oscar Lanzi Oct 13 '17 at 15:55
  • Yes they are. This may be anecdotal but there's a story about an early computer programmed to do geometry proofs. They asked it to prove the base angles of an isoceles triangle are congruent. The program Said since AB = AC and AC=AB and BC=BC then the two triangles ABC and ACB are congruent by Side-Side-Side and so angles B and C are congruent due to corresponding parts..... which is good story, although I highly doubt it is true. – fleablood Oct 13 '17 at 16:01
  • I suppose one can add an "orientation" factor to distinguish the two. If so we really need to point out that "mirror image" is 98.76 percent as strong. – fleablood Oct 13 '17 at 16:08
  • I'm giving +1 to the question. It shows legitimate and critical concern over an ambiguous point. Over which confusion is valid. – fleablood Oct 13 '17 at 17:18
  • Actually 3D mirror images ARE congruent. Congruence allows for mirror image. In advanced geometries cases and definitions for orientation can be made if needed but basic congruence is up to orientation. And 3D mirror images CAN be superimposed. You just need to flip them over across the 4th dimensional axis. – fleablood Oct 14 '17 at 00:43

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Yes obviously they are .because they just need to have equality with angle and side for triangles.so for this follow that method of congruence that will be helpfull. So (a,d) and (b,c) are the pairs you wanted. No problem with mirror i think.

Akash Mishra
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  • I wouldn't say it is "obvious". To be congruent they must more or less be the "same" or "able to fit on each other" and orientation is a notable difference. However it isn't a "significant" difference. Any thing about proving corresponding parts, or consequences (such as whether lines are parallel or whatever) will not matter if the two shapes have different orientations. – fleablood Oct 13 '17 at 17:12
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    Also note: In 3 dimensions. mirror images are also congruent. In 2D we can see that we can just pick up a triangle and flip it upside down (in the 3rd dimension). In 3D we can not. So if our intuition is "congruent means that we can move one to occupy the exact space of the other" that definition is incorrect unless we allow rotation through 1 more dimension than we have. Which we do. – fleablood Oct 13 '17 at 17:16
  • @fleablood Your Comment is the exact reason behind my question. Please check the updated question. – Sagar Upadhyay Oct 13 '17 at 21:25
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    It's a good question. But the answer is still Yes. Congruency holds over mirror image and superimposing allows for a third dimensional flip. – fleablood Oct 14 '17 at 00:45