Question
Which of the following measures would give you an infinite number of possible triangles?
26°, 42°, and 106°
34°, 63°, and 79°
62°, 34°, and 80°
three angles: 43°, 62°, and 75°
Answers
GPT-4o mini
To determine which set of angles can form an infinite number of triangles, we need to recall the triangle sum theorem, which states that the sum of the measures of the angles in a triangle must equal 180°. However, if the angles given are not valid in forming a triangle (i.e., they do not sum to 180°), we can assess what would happen in other scenarios.
Let’s evaluate each set of angles:
1. **26°, 42°, and 106°**
- Sum = 26 + 42 + 106 = 174° (not 180°)
2. **34°, 63°, and 79°**
- Sum = 34 + 63 + 79 = 176° (not 180°)
3. **62°, 34°, and 80°**
- Sum = 62 + 34 + 80 = 176° (not 180°)
4. **43°, 62°, and 75°**
- Sum = 43 + 62 + 75 = 180° (valid triangle)
From the above analysis, only the last set of angles sums to exactly 180°. The other three sets do not sum to 180°, which means they cannot form a triangle.
Hence, the answer is:
None of these measures give an infinite number of possible triangles; however, only the last set can construct a triangle. The first three can yield no triangles.
Let’s evaluate each set of angles:
1. **26°, 42°, and 106°**
- Sum = 26 + 42 + 106 = 174° (not 180°)
2. **34°, 63°, and 79°**
- Sum = 34 + 63 + 79 = 176° (not 180°)
3. **62°, 34°, and 80°**
- Sum = 62 + 34 + 80 = 176° (not 180°)
4. **43°, 62°, and 75°**
- Sum = 43 + 62 + 75 = 180° (valid triangle)
From the above analysis, only the last set of angles sums to exactly 180°. The other three sets do not sum to 180°, which means they cannot form a triangle.
Hence, the answer is:
None of these measures give an infinite number of possible triangles; however, only the last set can construct a triangle. The first three can yield no triangles.