Question

The triangle inequality theorem states that:
A. The sum of the length of $2$ sides of a triangle is equal to the third side of the triangle.
B. The sum of the length of $2$ sides of a triangle is less than the third side of the triangle.
C. The sum of the length of $2$ sides of a triangle is more than the third side of the triangle.
D. None of these.

Answer 1

Hint: For the given question first of all we will draw a triangle with sides$a,b,c$. Then we will give any desired value to the sides of the triangle. After that, we will apply the given condition in the triangle to find the answer. We will add any two sides of the triangle and put it equal to the third one as given in options. We will try this for all three options and see which one option is satisfied according to the example. Thus we will poof any of the given statements for triangle inequality theorem.

Complete step by step Answer:

Theorem: Triangle inequality theorem states that the sum of the length of $2$ sides of a triangle is greater than the third side of the triangle.
Let us prove this statement with the help of examples.
Example $1$ :
Consider the diagram:

In the above diagram three sides $a,b,c$are given.
Where:
 $
  a = 9m \\
  b = 8m \\
  c = 3m \\
 $
Now check this with the given statements.
We will add any two sides of the triangle and put it equal to the third side.
Condition$1$ : $AB + AC = BC$
$9 + 8 = 17 > 3$
It means
$AB + AC > BC$
Condition$2$: $AB + BC = AC$
$9 + 3 = 12 > 8$
It means
$AB + BC > AC$
Condition $3:$ $AC + BC = AB$
$8 + 3 = 11 > 9$
It means
$AC + BC > AB$
Thus we get that the sum of length of $2$ sides of a triangle is greater than the third side of the triangle.
Hence the correct answer is option C.

Note: For the given question first we have to consider any example and make a diagram of the triangle with sides $a,b,c$. Give some value to all three sides of the triangle, without doing this we cannot prove the theorem. We have to add any two sides of the triangle and put it equal to the third one, thus we get the correct statement.