Calculating the Possible Values of x in an Acute Angle Triangle
To find the possible values of x in an acute angle triangle with sides 10, 14, and x, we can use 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 third side.

Triangle Inequality Theorem
- For the sides 10, 14, and x to form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
- So, we have the following inequalities:
- 10 + 14 > x
- 10 + x > 14
- 14 + x > 10

Solving the Inequalities
- By solving the first inequality, we get: 24 > x
- By solving the second inequality, we get: x > 4
- By solving the third inequality, we get: x > -4

Combining the Inequalities
- Combining the inequalities, we find that x must be greater than 4 and less than 24.
- Therefore, the possible values of x are natural numbers between 5 and 23, inclusive.
- So, the total number of possible natural numbers for x is 19 (23-5+1 = 19).
By following the triangle inequality theorem and solving the inequalities, we determined that there are 19 possible natural numbers for x in the given acute angle triangle with sides 10, 14, and x.