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3 Responses
Good question. A reflection doesn’t change which sides are corresponding; it only reverses the orientation. In similarity, the correspondence is determined by matching equal angles. Once you fix ∠A ↔ ∠D and ∠B ↔ ∠E, the third angle forces ∠C ↔ ∠F. Then the corresponding sides are exactly the sides between matched angle-vertices: AB ↔ DE, AC ↔ DF, and BC ↔ EF. A mirror image just means that if you walk around ABC in one direction (say A→B→C), you have to walk around DEF in the opposite direction (D→E→F) to keep the angle matches aligned.
Since all three ratios AB/DE, AC/DF, and BC/EF come out equal to the same k, that’s solid confirmation that your pairing is consistent. The only caveat I can think of is in highly symmetric cases (like isosceles or equilateral triangles) where multiple pairings can accidentally produce the same ratios, but even then the angle-matching rule still guides you to the same correspondence. So, unless there’s some labeling quirk I’m not seeing, your matching looks correct.
You’re good! In similarity, the correspondence is determined by the equal angles, not by whether one triangle is a mirror image of the other. If ∠A = ∠D and ∠B = ∠E, then automatically ∠C = ∠F, so the vertex matching is A ↔ D, B ↔ E, C ↔ F. That means the corresponding sides are the ones opposite corresponding angles (or, equivalently, the sides between the matching angle pairs): AB ↔ DE, AC ↔ DF, and BC ↔ EF-exactly what you chose. A reflection just reverses the orientation (the direction you’d “walk” around the triangle), but it doesn’t change which angles line up, so it doesn’t change which sides correspond. Your pairing is fine. Hope this helps!
Mirroring doesn’t change which sides correspond in similarity-you match by equal angles (A↔D, B↔E, C↔F), so your AB↔DE, AC↔DF, BC↔EF pairing is peachy; it’s like the same triangle taking a mirror selfie-same features, just reversed. Think resizing and flipping a pizza slice (same crust, flipped direction-some folks even call the scale factor “negative” then); quick refresher: https://www.khanacademy.org/math/geometry/hs-geo-similarity/hs-geo-similar-triangles/a/similar-triangles-intro