Question

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

Explanation:

Step1: Check triangle inequality for side - side - side cases

The triangle inequality theorem states that for any triangle with side lengths \(a\), \(b\), and \(c\), \(a + b>c\), \(a + c>b\), and \(b + c>a\).
For sides \(4\) ft, \(8\) ft, and \(14\) ft: \(4+8 = 12<14\), so no triangle is formed.
For sides \(5\) in, \(12\) in, and \(14\) in: \(5 + 12=17>14\), \(5+14 = 19>12\), \(12 + 14=26>5\). By SSS (Side - Side - Side) congruence criterion, a unique triangle is formed.

Step2: Check angle - angle - angle cases

For three angles \(50^{\circ},50^{\circ},50^{\circ}\): \(50+50 + 50=150
eq180\), so no triangle is formed.
For three angles \(25^{\circ},65^{\circ},90^{\circ}\): \(25 + 65+90=180\). But by AAA (Angle - Angle - Angle) similarity criterion, infinitely many similar (not necessarily unique) triangles can be formed.

Answer:

three sides measuring \(5\) in., \(12\) in., and \(14\) in.