Final answer:
After applying the Triangle Inequality Theorem, we find that only Option A: 20, 21, and 22 can form a triangle. Options B, C, and D do not satisfy the theorem and therefore cannot form triangles.
Step-by-step explanation:
To determine which of the given measures can make a triangle, we can use the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Let's evaluate each option:
- Option A: 20, 21, and 22. These lengths satisfy the Triangle Inequality Theorem (20 + 21 > 22, 20 + 22 > 21, and 21 + 22 > 20), so they can make a triangle.
- Option B: 8, 80, and 80. Although the lengths of two sides are equal (80 and 80), the third side is too short (8) to satisfy the Triangle Inequality Theorem (8 + 80 > 80 is false), so they cannot make a triangle.
- Option C: 4, 40, 50. Again, the Triangle Inequality Theorem is not satisfied (4 + 40 < 50), so they cannot make a triangle.
- Option D: 3, 3, and 6. These lengths do not satisfy the Triangle Inequality Theorem (3 + 3 < 6), hence they cannot make a triangle.
Therefore, only Option A represents lengths that can form a triangle.