Triangle Inequality Theorem

Triangle Inequality Theorem

The sum of any two sides of a triangle must be greater than the third side.

For triangle with sides a, b, c:

• a + b > c
• a + c > b
• b + c > a

All three inequalities must be true!

Can These Sides Form a Triangle?

Test all three inequalities

Example 1: Valid Triangle

Sides: 3, 4, 5

Check:

• 3 + 4 > 5 → 7 > 5 ✓
• 3 + 5 > 4 → 8 > 4 ✓
• 4 + 5 > 3 → 9 > 3 ✓

All true → Yes, forms a triangle!

Example 2: NOT a Triangle

Sides: 2, 3, 6

Check:

• 2 + 3 > 6 → 5 > 6 ✗

One is false → No, cannot form a triangle!

Why? The two shorter sides (2 and 3) can't "reach" to close the triangle.

Example 3: Edge Case

Sides: 3, 4, 7

Check:

• 3 + 4 > 7 → 7 > 7 ✗

Equal, not greater → No triangle!

Must be strictly greater than, not equal to

Example 4: Large Numbers

Sides: 10, 15, 20

Quick check: Is 10 + 15 > 20?

• 25 > 20 ✓

Since smallest two sum > largest, all inequalities true

Yes, forms a triangle!

Shortcut Method

To quickly check:

1. Add the two smaller sides
2. Compare to the largest side
3. If sum > largest, it's a triangle

This works because: If the two smallest sum to more than the largest, the other inequalities are automatically true.

Example: Using Shortcut

Sides: 5, 8, 11

Smallest two: 5 + 8 = 13 Largest: 11

13 > 11 ✓ → Forms a triangle!

Finding Range of Third Side

Given two sides, find possible values for the third side

Rule: If sides are a and b, third side c must satisfy:

• |a - b| < c < a + b

In words:

• c must be less than a + b (sum)
• c must be greater than |a - b| (difference)

Example 1: Find Range

Two sides: 5 and 8

Find range for third side:

Lower bound: |5 - 8| = 3 Upper bound: 5 + 8 = 13

Range: 3 < c < 13

In words: Third side must be between 3 and 13 (not including 3 or 13)

Example 2: Possible Integers

Two sides: 4 and 7. What integer lengths work for third side?

Range: |4 - 7| < c < 4 + 7

• 3 < c < 11

Possible integers: 4, 5, 6, 7, 8, 9, 10

Answer: 7 possible integer lengths

Example 3: Large Sides

Two sides: 15 and 20

Range: |15 - 20| < c < 15 + 20

• 5 < c < 35

Third side between 5 and 35

Why This Works

Imagine walking from point A to point B:

Direct path: Length c

Detour through point C:

• Walk length a to C
• Walk length b from C to B
• Total: a + b

The direct path must be shorter than the detour!

So: c < a + b

Isosceles and Equilateral Cases

Isosceles triangle: Two equal sides

Example: Isosceles Triangle

Two sides are 6 and 6. Find range for third side.

Range: |6 - 6| < c < 6 + 6

• 0 < c < 12

So c can be any value from 0 to 12 (not including endpoints)

If c = 6: Equilateral triangle If c ≠ 6: Isosceles (but not equilateral)

Triangle Type from Sides

After confirming it's a triangle, classify it:

Acute: All angles < 90°

• If a² + b² > c² (for longest side c)

Right: One 90° angle

• If a² + b² = c² (Pythagorean)

Obtuse: One angle > 90°

• If a² + b² < c²

Example: Classify Triangle

Sides: 6, 8, 10

First, verify it's a triangle:

• 6 + 8 > 10 ✓

Classify:

• 6² + 8² = 36 + 64 = 100
• 10² = 100
• 6² + 8² = 10²

Right triangle! (3-4-5 scaled by 2)

Real-World Applications

Construction: Planning triangular structures

• Check if materials can form stable triangle

Navigation: Three-point location

• Triangle inequality ensures valid position

Engineering: Truss design

• Support beams must satisfy triangle inequality

Sports: Relay race paths

• Direct route vs. indirect routes

Practice