Question

which dimensions can create more than one triangle? three sides measuring 5 in., 12 in., and 13 in. three angles measuring 25°, 65°, and 90° three angles measuring 50°, 50°, and 50° three sides measuring 4 ft, 8 ft, and 10 ft

Explanation:

Step1: Check triangle - side conditions

By SSS (Side - Side - Side) congruence criterion, if three side lengths are given (\(a = 5\), \(b = 12\), \(c = 13\) and \(a'=4\), \(b' = 8\), \(c'=10\)), only one unique triangle can be formed. Because for a given set of three side lengths (satisfying the triangle inequality \(a + b>c\), \(a + c>b\), \(b + c>a\)), there is a unique triangle shape.

Step2: Check triangle - angle conditions

For three angles (\(\angle A=25^{\circ}\), \(\angle B = 65^{\circ}\), \(\angle C=90^{\circ}\)), since the sum of angles in a triangle is \(180^{\circ}\) (\(25 + 65+90=180\)), but the side lengths are not fixed. We can have similar triangles (same - shape, different - size) with the same set of angles. For three angles \(\angle A=\angle B=\angle C = 50^{\circ}\), \(50+50 + 50=150
eq180\), so it cannot form a triangle.

Answer:

three angles measuring \(25^{\circ}\), \(65^{\circ}\), and \(90^{\circ}\)