Triangle Inequality Theorem
key notes :
🟢 Triangle Inequality Theorem
🔺 Definition:
The sum of the lengths of any two sides of a triangle is always greater than the length of the third side. Formula:
If a triangle has sides a, b, c, then:
a + b > c
b + c > a
a + c > b
🔹 Purpose:
Helps to check whether three given lengths can form a triangle.
🔹 Example:
If sides are 3 cm, 4 cm, 8 cm → Can they form a triangle?
3 + 4 = 7 ❌ (not > 8)
✅ So, no triangle is possible.
🔹 Important Points:
⚠️ The sum of two sides must be strictly greater, not equal.
✅ Always check all three combinations.
🔄 Works for all types of triangles: scalene, isosceles, equilateral.
🌟 Mnemonic Tip:
“Any two sides together must be bigger than the third side” 👫➕👫 > 👬
🖌️ Visual Representation:
Draw a triangle and label sides a, b, c.
Show arrows with a + b > c, etc. to remember the theorem easily.
💡 Real-life Example:
Building a triangular roof 🏠
Connecting supports in a triangular frame 🏗️
Ensures structure is stable and possible.
Learn with an example
Can the sides of a triangle have lengths 1, 6, and 9?
First, put the three numbers in order from smallest to largest: a=1, b=6, and c=9.
Now check whether a+b>c. Since 1+6=7, it is false that 1+6>9. So, these are not the side lengths of a triangle.
Can the sides of a triangle have lengths 1, 2, and 3?
First, put the three numbers in order from smallest to largest: a=1, b=2, and c=3.
Now check whether a+b>c. Since 1+2=3, it is false that 1+2>3. So, these are not the side lengths of a triangle.
Can the sides of a triangle have lengths 3, 6, and 8?
First, put the three numbers in order from smallest to largest: a=3, b=6, and c=8.
Now check whether a+b>c. Since 3+6=9, it is true that 3+6>8. So, these are the side lengths of a triangle.
Can the sides of a triangle have lengths 3, 6, and 10?
First, put the three numbers in order from smallest to largest: a=3, b=6, and c=10.
Now check whether a+b>c. Since 3+6=9, it is false that 3+6>10. So, these are not the side lengths of a triangle.
Can the sides of a triangle have lengths 2, 3, and 5?
First, put the three numbers in order from smallest to largest: a=2, b=3, and c=5.
Now check whether a+b>c. Since 2+3=5, it is false that 2+3>5. So, these are not the side lengths of a triangle.
Can the sides of a triangle have lengths 2, 7, and 10?
First, put the three numbers in order from smallest to largest: a=2, b=7, and c=10.
Now check whether a+b>c. Since 2+7=9, it is false that 2+7>10. So, these are not the side lengths of a triangle.
Can the sides of a triangle have lengths 1, 10, and 10?
First, put the three numbers in order from smallest to largest: a=1, b=10, and c=10.
Now check whether a+b>c. Since 1+10=11, it is true that 1+10>10. So, these are the side lengths of a triangle.

Let’s practice!