Indirect Proof & the Triangle Inequality

This lesson pairs two ideas. The first is a proof strategy: an indirect proof, also called proof by contradiction, proves a statement by assuming the opposite of what you want to show and following that assumption until it crashes into something impossible. If the opposite cannot be true, the original statement must be.

The second is the Triangle Inequality: the sum of the lengths of any two sides of a triangle is greater than the length of the third side, $a + b > c$ . It is the rule that decides whether three lengths can close up into a triangle at all, and it also pins down the range of possible lengths for a missing third side.

How an indirect proof works

An indirect proof has three moves. First, assume the negation of the conclusion — the statement that covers every way the conclusion could fail. Second, reason from that assumption until you reach a contradiction of something known to be true. Third, conclude that the assumption was impossible, so the original statement holds.

The step students miss is writing the full negation. The opposite of $m\angle A > 40^\circ$ is not $m\angle A < 40^\circ$ — that skips the case where the angle equals $40^\circ$ . The complete negation is $m\angle A \leq 40^\circ$ . In general, the negation of greater than is less than or equal to, and the negation of less than is greater than or equal to.

Testing three lengths with the Triangle Inequality

Three lengths form a triangle only if every pair of sides adds to more than the remaining side. In practice you only need one check: add the two shortest lengths and compare with the longest. If the two shortest together beat the longest, the other two pairings pass automatically.

The comparison must be strictly greater. If the two shorter sides add up to exactly the longest side, like $3 + 4 = 7$ , the pieces lie flat in a straight line and no triangle forms. Greater than, not greater than or equal to. In the triangle below, each pair of sides must sum to more than the third for the figure to close up.

The range for a third side

If two sides of a triangle are known, the Triangle Inequality traps the third side between two bounds: it must be less than the sum of the known sides and greater than their difference. With sides $a$ and $b$ where $a > b$ , the third side $x$ satisfies $a - b < x < a + b$ .

Both bounds are strict — the third side can never equal the sum or the difference. So for sides of $6$ and $10$ , the third side is any length strictly between $4$ and $16$ .

Worked examples

Example 1: can three lengths form a triangle?

Can side lengths $4$ , $5$ , and $12$ form a triangle?

Add the two shortest lengths

4 + 5 = 9

Compare with the longest length

9 < 12

The two shorter sides cannot reach past the longest side, so no triangle forms

Answer: No — $4 + 5$ is not greater than $12$

Example 2: the range for a third side

Two sides of a triangle measure $6$ and $10$ . What are the possible lengths of the third side?

Upper bound: less than the sum

x < 6 + 10 = 16

Lower bound: greater than the difference

x > 10 - 6 = 4

Combine the bounds

4 < x < 16

Answer: $4 < x < 16$

Example 3: starting an indirect proof

Set up the first step of an indirect proof of the statement “ $m\angle A > 40^\circ$ .”

An indirect proof begins by assuming the negation of the conclusion

The negation must cover every failing case: less than and equal to

m\angle A < 40^\circ \text{ or } m\angle A = 40^\circ

Write the two cases as one statement

m\angle A \leq 40^\circ

Answer: Assume $m\angle A \leq 40^\circ$

Try one yourself

Common questions

Why do I only check the two shortest sides?

Any sum involving the longest side is automatically bigger than the leftover side, so those pairings never fail. The only pairing at risk is the two shortest sides against the longest — if that one passes, all three pass.

What if the two shorter sides add up to exactly the longest side?

No triangle. The three pieces flatten into a straight segment instead of closing up. The Triangle Inequality requires strictly greater than, so a set like $3, 4, 7$ fails.

Why does the negation of “greater than” include “equal to”?

Because the negation must cover every case where the original statement is false. The statement $m\angle A > 40^\circ$ is false both when the angle is smaller than $40^\circ$ and when it equals $40^\circ$ , so the assumption has to be $m\angle A \leq 40^\circ$ .

Can the third side equal the difference of the other two?

No. Both bounds are strict: the third side must be strictly greater than the difference and strictly less than the sum. At either bound the figure collapses flat.

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