Question
Sore into categories, unique triangles, not unique triangles(many triangles), and no triangles(impossible)
60°, 60° 60°
4 cm, 5 cm, 6 cm
8m, 1m, 8m
30°, 70°, 80°,
90°, 40°, 40°
10 cm, 12 cm, 22 cm
60°, 60° 60°
4 cm, 5 cm, 6 cm
8m, 1m, 8m
30°, 70°, 80°,
90°, 40°, 40°
10 cm, 12 cm, 22 cm
Answers
GPT-4o mini
To determine how the given sets of angles or side lengths can form triangles, we can classify them into three categories: unique triangles, not unique triangles (many triangles), and no triangles (impossible).
### Unique Triangles
1. **60°, 60°, 60°**: This forms an equilateral triangle. It is unique since all angles are equal (and thus all sides are equal).
2. **30°, 70°, 80°**: This set of angles is valid and sums to 180°. It forms a unique triangle since all angles are different.
3. **4 cm, 5 cm, 6 cm**: This set of sides satisfies the triangle inequality theorem (4 + 5 > 6, 5 + 6 > 4, 4 + 6 > 5), forming a unique triangle.
### Not Unique Triangles
1. **8m, 1m, 8m**: This set of sides can form an isosceles triangle. Because of the two equal sides (8m), there is more than one possible triangle configuration (within the constraints of the measure).
2. **90°, 40°, 40°**: Since one angle is 90° and the other two are equal, this also implies multiple configurations must hold (isosceles right triangle).
### No Triangles (Impossible)
1. **10 cm, 12 cm, 22 cm**: This set of sides does not satisfy the triangle inequality theorem (10 + 12 = 22, which does not satisfy the strict inequality), therefore, it is impossible to form a triangle.
### Summary:
- **Unique Triangles**: 60°, 60°, 60°; 30°, 70°, 80°; 4 cm, 5 cm, 6 cm.
- **Not Unique Triangles**: 8m, 1m, 8m; 90°, 40°, 40°.
- **No Triangles (Impossible)**: 10 cm, 12 cm, 22 cm.
### Unique Triangles
1. **60°, 60°, 60°**: This forms an equilateral triangle. It is unique since all angles are equal (and thus all sides are equal).
2. **30°, 70°, 80°**: This set of angles is valid and sums to 180°. It forms a unique triangle since all angles are different.
3. **4 cm, 5 cm, 6 cm**: This set of sides satisfies the triangle inequality theorem (4 + 5 > 6, 5 + 6 > 4, 4 + 6 > 5), forming a unique triangle.
### Not Unique Triangles
1. **8m, 1m, 8m**: This set of sides can form an isosceles triangle. Because of the two equal sides (8m), there is more than one possible triangle configuration (within the constraints of the measure).
2. **90°, 40°, 40°**: Since one angle is 90° and the other two are equal, this also implies multiple configurations must hold (isosceles right triangle).
### No Triangles (Impossible)
1. **10 cm, 12 cm, 22 cm**: This set of sides does not satisfy the triangle inequality theorem (10 + 12 = 22, which does not satisfy the strict inequality), therefore, it is impossible to form a triangle.
### Summary:
- **Unique Triangles**: 60°, 60°, 60°; 30°, 70°, 80°; 4 cm, 5 cm, 6 cm.
- **Not Unique Triangles**: 8m, 1m, 8m; 90°, 40°, 40°.
- **No Triangles (Impossible)**: 10 cm, 12 cm, 22 cm.