Which measurements could create more than one triangle? Explain how you found your answer, show your work.
A.
A triangle with sides measuring 6 inches, 8 inches, and 10 inches

B.
A triangle with sides measuring 5 cm and 10 cm and an included angle measuring 40°
C.
A triangle with angles measuring 75°, 60°, and 45°

D.
A triangle with sides measuring 4 inches, 8 inches, and 15 inches
To determine which measurements can create more than one triangle, we need to apply the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
Let's examine each option:
A. Triangle with sides measuring 6 inches, 8 inches, and 10 inches:
Using the triangle inequality theorem, we check if the sum of any two sides is greater than the remaining side.
- 6 + 8 > 10 (14 > 10) - This condition is met.
- 6 + 10 > 8 (16 > 8) - This condition is met.
- 8 + 10 > 6 (18 > 6) - This condition is met.
Since all three conditions are met, the given measurements can create a triangle.
B. Triangle with sides measuring 5 cm and 10 cm and an included angle measuring 40°:
To determine if this measurement can create more than one triangle, we need to use the law of sines.
The law of sines states that in any triangle, the ratio of the length of a side to the sine of the angle opposite that side is constant.
Using the law of sines, we can find the possible lengths of the third side using the given angle and the ratio of the lengths of the sides:
sin(40°) / 5 = sin(180° - 40° - x) / x
where x represents the length of the third side.
By solving this equation, we can find the possible values of x.
C. Triangle with angles measuring 75°, 60°, and 45°:
The sum of the angles in a triangle must be 180°, so let's check if that is the case:
75° + 60° + 45° = 180°
Since the sum of the angles is 180°, the given angles can create a triangle.
D. Triangle with sides measuring 4 inches, 8 inches, and 15 inches:
Using the triangle inequality theorem, let's check if the sum of any two sides is greater than the remaining side:
- 4 + 8 > 15 (12 > 15) - This condition is not met.
Since one of the conditions is not met, the given measurements cannot create a triangle.
In summary:
- Options A and C have measurements that can create a triangle.
- Option B requires further analysis using the law of sines.
- Option D cannot create a triangle.