The ABC Toy Company is creating two similar pieces for a board game, as shown below. How could the toy maker confirm that the pieces are similar using the definition of similarity in terms of similarity transformations? triangles GHI and JKL Verify ∠G ≅ ∠J and ∠I ≅ ∠L by reflection. Verify segment GH ≅ segment JK and segment GI ≅ segment JL by reflection. Verify ∠G ≅ ∠J and ∠I ≅ ∠L by translation. Verify segment GH ≅ segment JK and segment GI ≅ segment JL by translation.
To confirm that the triangles GHI and JKL are similar using the definition of similarity in terms of similarity transformations, we can verify the following conditions:
1. Verify ∠G ≅ ∠J and ∠I ≅ ∠L:
By reflecting triangle GHI and triangle JKL, we can check if the corresponding angles are congruent. If the reflected images of ∠G and ∠I coincide with ∠J and ∠L respectively, then it confirms that ∠G ≅ ∠J and ∠I ≅ ∠L.
2. Verify segment GH ≅ segment JK and segment GI ≅ segment JL:
By reflecting triangle GHI and triangle JKL, we can check if the corresponding sides are congruent. If the reflected images of segment GH and segment GI coincide with segment JK and segment JL respectively, then it confirms that segment GH ≅ segment JK and segment GI ≅ segment JL.
Alternatively, the toy maker can also verify these conditions using translation instead of reflection. By translating triangle GHI and triangle JKL, they can check if the corresponding angles and sides overlap. If the translated images of ∠G and ∠I coincide with ∠J and ∠L respectively, and the translated images of segment GH and segment GI coincide with segment JK and segment JL respectively, then it confirms the similarity of the triangles.
To confirm that triangles GHI and JKL are similar using the definition of similarity in terms of similarity transformations, we can follow the given steps:
1. Verify ∠G ≅ ∠J and ∠I ≅ ∠L by reflection:
- Reflect triangle GHI over a line of reflection to create a new triangle G'HI.
- Reflect triangle JKL over the same line of reflection to create a new triangle J'KL.
- Check if ∠G' ≅ ∠J' and ∠I ≅ ∠L.
2. Verify segment GH ≅ segment JK and segment GI ≅ segment JL by reflection:
- Reflect triangle GHI over a line of reflection to create a new triangle G'HI.
- Reflect triangle JKL over the same line of reflection to create a new triangle J'KL.
- Check if segment G'H ≅ segment J'K and segment G'I ≅ segment J'L.
3. Verify ∠G ≅ ∠J and ∠I ≅ ∠L by translation:
- Translate triangle GHI horizontally or vertically to create a new triangle G''HI.
- Translate triangle JKL by the same distance and direction to create a new triangle J''KL.
- Check if ∠G'' ≅ ∠J'' and ∠I ≅ ∠L.
4. Verify segment GH ≅ segment JK and segment GI ≅ segment JL by translation:
- Translate triangle GHI horizontally or vertically to create a new triangle G''HI.
- Translate triangle JKL by the same distance and direction to create a new triangle J''KL.
- Check if segment G''H ≅ segment J''K and segment G''I ≅ segment J''L.
By performing these steps, the toy maker can verify whether the triangles GHI and JKL are similar using the definition of similarity in terms of similarity transformations.