The Smaller Angle Has the Opposite Side:

Introduction:
In a triangle, the sum of all three angles is always 180 degrees. If two angles of a triangle are unequal, it means that one angle is larger than the other. In such a case, the side opposite to the larger angle will be longer than the side opposite to the smaller angle. This relationship can be explained using the concept of the triangle inequality theorem.

Triangle Inequality Theorem:
The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This theorem helps us understand the relationship between the angles and sides of a triangle.

The Smaller Angle and Its Opposite Side:
When two angles of a triangle are unequal, let's say angle A and angle B, we can assume that angle A is smaller than angle B. Now, let's consider the sides opposite to these angles, denoted as side a and side b respectively.

According to the triangle inequality theorem, the sum of the lengths of side a and side b must be greater than the length of the third side, denoted as side c. Mathematically, this can be represented as:

a + b > c

Since angle B is larger than angle A, it implies that side b is longer than side a. Therefore, we can rewrite the inequality as:

a + b > c
a + b > a (since side b > side a)
b > 0

As we can see, the inequality b > 0 holds true, indicating that side b is indeed longer than side a.

Conclusion:
In summary, if two angles of a triangle are unequal, the smaller angle will have the side opposite to it as the shorter side. This relationship is based on the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Length of side will be proportional to the angle opposite to it