To determine which measurements could create more than one triangle, we can use the Triangle Inequality Theorem, which states that for any triangle:
The sum of the lengths of any two sides must be greater than the length of the third side.
Let's analyze the given options:
A. A triangle with sides measuring 9 inches, 12 inches, and 15 inches
Here, 9 + 12 = 21, which is greater than 15. So, it forms a valid triangle.
B. A right triangle with acute angles measuring 20° and 70°
The sum of the angles in a triangle must be 180°. 20° + 70° = 90°, so this doesn't form a triangle.
C. A triangle with angles measuring 60°, 70°, and 80°
The sum of the angles in a triangle must be 180°. 60° + 70° + 80° = 210°, so this doesn't form a triangle.
D. A triangle with sides measuring 20 cm and 35 cm and an included angle measuring 45°
You can use the Law of Sines or the Law of Cosines to determine if this forms a triangle. Let's use the Law of Cosines:
c^2 = a^2 + b^2 - 2ab * cos(C)
c^2 = 20^2 + 35^2 - 2 * 20 * 35 * cos(45°)
c^2 = 400 + 1225 - 1400 * 0.7071 ≈ 162.5
Taking the square root of 162.5, we get c ≈ 12.74 cm.
Since all sides are positive and the angle measures are valid, this forms a valid triangle.
So, the measurements that could create more than one triangle are A and D.