Proof: A triangle is acute angled if each angle is less than the sum of the other two.

Introduction:
In geometry, a triangle is a polygon with three sides and three angles. The sum of the interior angles of a triangle is always 180 degrees. An acute triangle is a triangle in which all three angles are less than 90 degrees. In this proof, we will show that if each angle of a triangle is less than the sum of the other two angles, then the triangle is acute angled.

Proof:

Step 1: Let's consider a triangle ABC with angles A, B, and C.

Step 2: Assume, without loss of generality, that angle A is the largest angle.

Step 3: According to the given condition, angle A is less than the sum of angles B and C.

Step 4: Suppose angle A is greater than or equal to 90 degrees.

Step 5: In this case, the sum of angles B and C must be less than or equal to 90 degrees since angle A is the largest angle.

Step 6: However, if the sum of angles B and C is less than or equal to 90 degrees, then the sum of all three angles (A + B + C) would also be less than or equal to 180 degrees.

Step 7: But we know that the sum of interior angles of a triangle is always 180 degrees.

Step 8: This is a contradiction, as the sum of the angles cannot be both less than or equal to 180 degrees and less than or equal to 90 degrees.

Step 9: Therefore, our assumption that angle A is greater than or equal to 90 degrees must be false.

Step 10: Hence, angle A must be less than 90 degrees.

Step 11: Similarly, we can prove that angles B and C are also less than 90 degrees.

Conclusion:
From the above proof, we can conclude that if each angle of a triangle is less than the sum of the other two angles, then the triangle is acute angled.