Question

question 13 of 15
which measurements could create more than one triangle?
a. a triangle with sides measuring 50 mm and 35 mm and a nonincluded angle measuring 25°
b. a triangle with sides measuring 5 cm, 10 cm, and 15 cm
c. a triangle with sides measuring 5 inches, 7 inches, and 9 inches
d. a triangle with sides measuring 8 cm and 12 cm and an included angle measuring 85°

Explanation:

Step1: Check triangle - side - side - angle (SSA) condition

For a triangle with two sides and a non - included angle (SSA), the ambiguous case (more than one triangle) can occur.

Step2: Analyze each option

• Option B:

Using the triangle inequality theorem \(a + b>c\), \(a + c>b\), \(b + c>a\). For \(a = 5\), \(b = 10\), \(c = 15\), \(5+10=15\), which does not satisfy the triangle inequality theorem. So, no triangle can be formed.

• Option C:

For three given side lengths \(a = 5\), \(b = 7\), \(c = 9\), by the SSS (side - side - side) congruence criterion, exactly one triangle can be formed.

• Option D:

For a triangle with two sides and an included angle (SAS - side - angle - side), by the SAS congruence criterion, exactly one triangle can be formed.

• Option A:

When we have two sides and a non - included angle (SSA), there is a possibility of the ambiguous case (more than one triangle).

Answer:

A. A triangle with sides measuring 50 mm and 35 mm and a nonincluded angle measuring \(25^{\circ}\)