Which of the following statements is FALSE?

Correct option is C

Solution:

A) In any triangle, the sum of any two sides of the triangle is greater than the length of the third side: This is a true statement, known as the triangle inequality theorem. 

B) The greatest side of any triangle makes acute angles with each of the other sides: 

This is generally a true statement. If a triangle's longest side is subtended by an acute angle,

then the other two angles must also be acute.

C) In any triangle, the sum of any two angles of the triangle is greater than the measure of the third angle: 

This is FALSE. The sum of the measures of any two angles in a triangle is greater than the measure of the third angle,

but it is not necessarily greater than all other angles. 

D) In any triangle, the difference between the lengths of any two sides of a triangle is smaller than the length of the third side:

This is a true statement, as it's part of the triangle inequality theorem.