Question
In which pair of triangles is Triangle E F G is congruent to triangle R T S?
For triangles G E F and S R T, sides E G and S T are congruent, sides G F and S R are congruent, and sides E F and R T are congruent.
For triangles G F E and S T R, sides E G and T R are congruent, sides G F and T S are congruent, and sides E F and S R are congruent.
For triangles G F E and R S T, sides E G and R S are congruent, sides G F and R T are congruent, and sides E F and T S are congruent.
For triangles E G F and T S R, sides E G And S R are congruent, sides G F and S T are congruent, and sides E F and T R are congruent.
For triangles G E F and S R T, sides E G and S T are congruent, sides G F and S R are congruent, and sides E F and R T are congruent.
For triangles G F E and S T R, sides E G and T R are congruent, sides G F and T S are congruent, and sides E F and S R are congruent.
For triangles G F E and R S T, sides E G and R S are congruent, sides G F and R T are congruent, and sides E F and T S are congruent.
For triangles E G F and T S R, sides E G And S R are congruent, sides G F and S T are congruent, and sides E F and T R are congruent.
Answers
GPT-4o mini
To determine which pair of triangles shows that Triangle EFG is congruent to Triangle RTS, we need to identify the triangles where corresponding sides are congruent.
1. In the first pair (triangles GEF and SRT), it is stated that:
- \( EG \cong ST \)
- \( GF \cong SR \)
- \( EF \cong RT \)
This shows that all three corresponding sides are congruent, which means these two triangles are congruent.
2. In the second pair (triangles GFE and STR), it is stated that:
- \( EG \cong TR \)
- \( GF \cong TS \)
- \( EF \cong SR \)
This does not match the corresponding sides for triangle congruency directly.
3. In the third pair (triangles GFE and RST), it is stated that:
- \( EG \cong RS \)
- \( GF \cong RT \)
- \( EF \cong TS \)
This does not match the original triangles either.
4. Lastly, in the fourth pair (triangles EGF and TSR), it is stated that:
- \( EG \cong SR \)
- \( GF \cong ST \)
- \( EF \cong TR \)
This also does not correctly match the corresponding sides.
The only pair where all side relationships correctly correspond to the assumption that Triangle EFG is congruent to Triangle RTS is the first pair of triangles (G E F and S R T).
Therefore, the answer is the first pair: **For triangles G E F and S R T, sides E G and S T are congruent, sides G F and S R are congruent, and sides E F and R T are congruent.**
1. In the first pair (triangles GEF and SRT), it is stated that:
- \( EG \cong ST \)
- \( GF \cong SR \)
- \( EF \cong RT \)
This shows that all three corresponding sides are congruent, which means these two triangles are congruent.
2. In the second pair (triangles GFE and STR), it is stated that:
- \( EG \cong TR \)
- \( GF \cong TS \)
- \( EF \cong SR \)
This does not match the corresponding sides for triangle congruency directly.
3. In the third pair (triangles GFE and RST), it is stated that:
- \( EG \cong RS \)
- \( GF \cong RT \)
- \( EF \cong TS \)
This does not match the original triangles either.
4. Lastly, in the fourth pair (triangles EGF and TSR), it is stated that:
- \( EG \cong SR \)
- \( GF \cong ST \)
- \( EF \cong TR \)
This also does not correctly match the corresponding sides.
The only pair where all side relationships correctly correspond to the assumption that Triangle EFG is congruent to Triangle RTS is the first pair of triangles (G E F and S R T).
Therefore, the answer is the first pair: **For triangles G E F and S R T, sides E G and S T are congruent, sides G F and S R are congruent, and sides E F and R T are congruent.**