Question

Which dimensions can create only one unique triangle?
three angles measuring 25 degrees, 65 degrees, and 90 degrees
three angles measuring 50 degrees, 50 degrees, and 50 degrees
three sides measuring 5 in., 12 in., and 14 in.
three sides measuring 4 ft, 8 ft, and 14 ft

Answer 1

Only the three sides 5 in., 12 in., and 14 in.

Reasons:
- Angles 25°, 65°, 90° (AAA) determine shape only up to similarity — infinitely many (scaled) triangles, not a single unique congruent triangle.
- Angles 50°, 50°, 50° sum to 150° ≠ 180°, so no triangle.
- Sides 5, 12, 14 satisfy the triangle inequalities, and SSS determines a unique triangle (up to congruence).
- Sides 4, 8, 14 fail the triangle inequality (4 + 8 = 12 ≤ 14), so no triangle.