5.2 Triangle inequalities and indirect proofs

Triangle Inequalities

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Triangle Inequality Theorem Application

The Triangle Inequality Theorem gives you a quick test for whether three side lengths can actually form a triangle. The rule: the sum of any two sides must be strictly greater than the third side.

For a triangle with sides $a$, $b$, and $c$, all three of these must be true:

• $a + b > c$
• $b + c > a$
• $c + a > b$

If even one of these fails, you can't build a triangle with those lengths.

There's an equivalent way to think about this using differences: the third side must be greater than the absolute difference of the other two.

• $|a - b| < c$
• $|b - c| < a$
• $|c - a| < b$

Example: Can sides 5, 7, and 10 form a triangle?

1. $5 + 7 = 12 > 10$ ✓
2. $7 + 10 = 17 > 5$ ✓
3. $5 + 10 = 15 > 7$ ✓

All three pass, so yes, these lengths form a valid triangle. Now try 2, 3, and 6: $2 + 3 = 5$, which is not greater than 6, so no triangle is possible.

Third Side Range in Triangles

When you know two sides and need to find the possible values for the third, the Triangle Inequality Theorem gives you a range. If the two known sides are $a$ and $b$, the third side $x$ must satisfy:

$|a - b| < x < a + b$

The third side has to be greater than the difference and less than the sum of the two known sides.

Example: For a triangle with sides 5 and 8, find the range of the third side $x$:

1. Compute the difference: $|5 - 8| = 3$

2. Compute the sum: $5 + 8 = 13$

3. Write the inequality: $3 < x < 13$

So $x$ can be any length between 3 and 13, but not equal to either endpoint. At exactly 3 or 13, the three "sides" would collapse into a straight line, not a triangle.

Indirect Proofs and Triangle Relationships

Indirect Proofs for Triangles

An indirect proof (also called proof by contradiction) works by assuming the opposite of what you want to prove, then showing that assumption leads to something impossible. Once you hit a contradiction, the original statement must be true.

Here are the steps:

1. Assume the opposite of the statement you want to prove.
2. Reason logically from that assumption, using definitions, theorems, and given information.
3. Arrive at a contradiction with a known fact, a given, or a previously proven theorem.
4. Conclude that the original statement is true, since the opposite led to a contradiction.

Example: Suppose you need to prove that a triangle does not have two obtuse angles. Assume the opposite: that it does have two obtuse angles. Each obtuse angle is greater than $90°$, so their sum alone exceeds $180°$. But the angle sum of any triangle equals exactly $180°$, leaving no room for a third angle. That's a contradiction, so a triangle cannot have two obtuse angles.

Indirect proofs come up often when you need to show that certain triangle configurations are impossible or that a particular side/angle relationship must hold.

Angle-Side Relationships in Triangles

Angles and sides in a triangle are directly linked by size: the longest side is always opposite the largest angle, and the shortest side is always opposite the smallest angle. This works in reverse too. If you know which angle is biggest, you know which side is longest.

A few key relationships to keep straight:

• If two sides are congruent (isosceles triangle), the angles opposite those sides are also congruent.
• If two angles are congruent, the sides opposite those angles are also congruent.
• The sum of all three interior angles is always $180°$.

For right triangles specifically, the Pythagorean Theorem connects the sides:

$a^2 + b^2 = c^2$

Here $a$ and $b$ are the legs and $c$ is the hypotenuse (the side opposite the right angle, which is always the longest side). This relationship only applies to right triangles, but you can use it alongside the angle-side relationships to solve for unknown measurements.