The Triangle Inequality Theorem

The Triangle Inequality Theorem tells us important rules about triangles. For any triangle with sides that are $a$, $b$, and $c$, these rules must always be true:

1. $a + b > c$
2. $a + c > b$
3. $b + c > a$

These rules work for all kinds of triangles. This includes:

• Scalene triangles (with all sides different),
• Isosceles triangles (with two equal sides), and
• Equilateral triangles (with all sides equal).

1. Scalene Triangles

In scalene triangles, every side has a different length. The Triangle Inequality Theorem helps us check that the longest side is shorter than the sum of the other two sides.

For example, imagine we have a scalene triangle with sides that measure 4, 5, and 7. Let’s see if the rules hold:

• $4 + 5 = 9 > 7$ (This is correct!)
• $4 + 7 = 11 > 5$ (This is also correct!)
• $5 + 7 = 12 > 4$ (This is still correct!)

All the rules are satisfied!

2. Isosceles Triangles

In isosceles triangles, two sides are the same length. Let’s say the two equal sides are $x$ and the base is $y$. We can still use the Triangle Inequality Theorem:

• $x + x > y$ (The equal sides must be longer than the base.)
• $x + y > x$ (This is always true because $y$ is positive.)
• $y + x > x$ (This is also always true as $y$ is positive.)

So, this shows that an isosceles triangle can exist.

3. Equilateral Triangles

In equilateral triangles, all sides are equal, which we call $s$. The inequalities become:

• $s + s > s$ (This means $2s > s$, which is always true!)

Since this is always true, we know the rules of the Triangle Inequality Theorem hold for equilateral triangles too.

Conclusion

The Triangle Inequality Theorem is a key idea that applies to all types of triangles. It helps us understand the conditions needed for a triangle to exist. This keeps our geometric shapes strong and well-structured!