In Triangle DEF, DE = 11cm and EF = 9 cm. For which measure of angle D is it possible to draw two scalene triangles.

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To determine for which measure of angle D it is possible to draw two scalene triangles, we need to consider the Triangle Inequality Theorem.

The Triangle Inequality Theorem states that for a triangle with side lengths a, b, and c, the sum of the lengths of any two sides must be greater than the length of the third side.

In this case, we are given DE = 11 cm and EF = 9 cm. To create two scalene triangles, we need two different lengths for side DE and EF.

Let's consider the following scenarios:

1. If angle D is acute or less than 90 degrees:
- In this case, we can draw a triangle with side lengths DE = 11 cm, EF = 9 cm, and any length for side DF greater than the difference between 11 cm and 9 cm. For example, if DF = 3 cm, we obtain a valid scalene triangle.
- For the second triangle, we can draw a triangle with side lengths DE = 11 cm, EF = 9 cm, and any length for side DF that is less than the sum of 11 cm and 9 cm but greater than the difference between them. For example, if DF = 15 cm, we obtain a valid scalene triangle.

2. If angle D is obtuse or greater than 90 degrees:
- In this case, we can still draw a triangle with side lengths DE = 11 cm, EF = 9 cm, and any length for side DF greater than the difference between 11 cm and 9 cm. For example, if DF = 3 cm, we obtain a valid scalene triangle.
- However, it is not possible to draw a second scalene triangle in this case because the sum of DE and EF is not greater than the length of the third side DF. This violates the Triangle Inequality Theorem.

In conclusion, it is possible to draw two scalene triangles if angle D is acute or less than 90 degrees.