Question

question 1 of 5
which measurements could create more than one triangle?
a. a triangle with sides measuring 6 inches, 8 inches, and 10 inches
b. a triangle with sides measuring 4 inches, 8 inches, and 15 inches
c. a triangle with angles measuring 75°, 60°, and 45°
d. a triangle with sides measuring 5 cm and 10 cm and an included angle measuring 40°

Explanation:

Step1: Check triangle with sides 6,8,10

By SSS (Side - Side - Side) congruence, a triangle with sides 6,8,10 is unique.

Step2: Check triangle with sides 4,8,15

Since \(4 + 8=12<15\), a triangle with sides 4,8,15 does not exist.

Step3: Check triangle with angles \(75^{\circ},60^{\circ},45^{\circ}\)

By AA (Angle - Angle) similarity, triangles with angles \(75^{\circ},60^{\circ},45^{\circ}\) are similar but not necessarily congruent. So, there can be infinitely many triangles with these angles (different side lengths but same angle measures).

Step4: Check triangle with sides 5,10 and included angle \(40^{\circ}\)

By SAS (Side - Angle - Side) congruence, a triangle with sides 5,10 and included angle \(40^{\circ}\) is unique.

Answer:

C. A triangle with angles measuring \(75^{\circ},60^{\circ}\), and \(45^{\circ}\)