Math Core

Can you build a triangle out of sticks that are $3$ , $4$ and $8$ inches long? Try it and you'll find the two short sticks can't reach each other. Not every set of three lengths makes a triangle, and inside any triangle the sizes of the sides and angles are tied together. This lesson covers the inequalities that control what triangles are possible.

Bigger sides face bigger angles

Picture a triangle where one angle is very wide. The side across from that angle has to stretch to connect the other two sides, so it is long. A narrow angle, on the other hand, pinches the opposite side short.

Side and angle relationships in one triangle

In any triangle:

If one side is longer than another, the angle opposite the longer side is larger.
If one angle is larger than another, the side opposite the larger angle is longer.

So the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle.

"Opposite" means the side that does not touch the angle. In $\triangle ABC$ , the side opposite $\angle A$ is $\overline{BC}$ , the side opposite $\angle B$ is $\overline{AC}$ , and the side opposite $\angle C$ is $\overline{AB}$ .

Worked example: Ordering the angles

In $\triangle ABC$ , $AB = 9$ , $BC = 12$ and $AC = 7$ . List the angles from smallest to largest.

Solution. Match each side with the angle across from it:

Side	Length	Opposite angle
$\overline{AC}$	$7$	$\angle B$
$\overline{AB}$	$9$	$\angle C$
$\overline{BC}$	$12$	$\angle A$

The sides in increasing order are $AC$ , $AB$ , $BC$ , so the angles in increasing order are $\angle B$ , $\angle C$ , $\angle A$ .

Worked example: Ordering the sides

In $\triangle ABC$ , $m\angle A = 50^\circ$ and $m\angle B = 72^\circ$ . List the sides from shortest to longest.

Solution. First find the third angle: $m\angle C = 180^\circ - 50^\circ - 72^\circ = 58^\circ$ .

The angles in increasing order are $\angle A$ ( $50^\circ$ ), $\angle C$ ( $58^\circ$ ), $\angle B$ ( $72^\circ$ ). Their opposite sides are $\overline{BC}$ , $\overline{AB}$ and $\overline{AC}$ . So

BC < AB < AC.

Common mistake

The most common error is pairing an angle with a side that touches it. $\overline{AB}$ touches both $\angle A$ and $\angle B$ , so it can't be opposite either one. It's opposite $\angle C$ , the one vertex letter it doesn't contain. A quick check: the side opposite an angle never uses that angle's letter.

The Triangle Inequality Theorem

Back to the sticks. Lay the $8$ -inch stick flat. The $3$ -inch and $4$ -inch sticks are hinged at its ends, so their free ends swing along circles. Even stretched flat along the base, they reach only $3 + 4 = 7$ inches, which is less than $8$ . They can never meet.

Sides of length 3 and 4 swing on circles (dashed) around the ends of a side of length 8. The circles never meet, so no triangle forms.

If the two short sides added up to exactly $8$ , they would lie flat on top of the base, which is a segment, not a triangle. To get a real triangle, the two shorter sides must add up to more than the third.

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side. For a triangle with sides $a$ , $b$ and $c$ :

a + b > c, \qquad a + c > b, \qquad b + c > a.

You don't have to check all three inequalities. If the two shortest sides add up to more than the longest side, the other two inequalities are automatically true.

Worked example: Can these lengths form a triangle?

Decide whether each set of lengths can be the sides of a triangle.

$5$ , $7$ , $13$
$4$ , $6$ , $9$
$6$ , $6$ , $12$

Solution. Add the two shortest and compare with the longest.

$5 + 7 = 12$ , and $12 < 13$ . No triangle.
$4 + 6 = 10$ , and $10 > 9$ . Yes, a triangle is possible.
$6 + 6 = 12$ , which is not greater than $12$ . No triangle (the sides would lie flat).

Finding the possible third side

If you know two sides of a triangle, the Triangle Inequality traps the third side between two values.

Range for the third side

If two sides of a triangle have lengths $a$ and $b$ with $a \ge b$ , the third side $x$ satisfies

a - b < x < a + b.

Worked example: The range of the third side

Two sides of a triangle are $6$ cm and $10$ cm long. What lengths are possible for the third side? How many of them are whole numbers?

Solution. The third side must be less than the sum and greater than the difference:

10 - 6 < x < 10 + 6 \quad\Longrightarrow\quad 4 < x < 16.

The whole-number lengths are $5, 6, 7, \dots, 15$ . That's $15 - 5 + 1 = 11$ possible lengths. Notice that $4$ and $16$ themselves are not allowed.

Tip

To check the lower bound, try it: with sides $10$ and $6$ , a third side of $3$ fails because $3 + 6 = 9$ is not more than $10$ . The short sides must beat the long side, so the third side has to exceed $10 - 6 = 4$ .

Comparing two triangles: the Hinge Theorem

Open a door a little, then open it wide. The door and its frame don't change length, but the gap between the door's edge and the frame grows as the hinge angle grows. Triangles behave the same way.

Two pairs of equal sides. The larger included angle (100° at D) is opposite the longer third side, EF.

Hinge Theorem and its converse

Suppose two sides of one triangle are congruent to two sides of another triangle.

Hinge Theorem: If the included angle of the first is larger, then the third side of the first is longer.
Converse: If the third side of the first is longer, then its included angle is larger.

In the figure, $AB = DE$ and $AC = DF$ , and $m\angle D > m\angle A$ , so $EF > BC$ .

Practice

Practice 1

Which set of lengths can form a triangle?

Practice 2

In $\triangle PQR$ , $PQ = 14$ , $QR = 6$ and $PR = 11$ . Which angle is the largest?

Practice 3

In $\triangle ABC$ , $m\angle A = 35^\circ$ and $m\angle B = 80^\circ$ . Which list orders the sides from shortest to longest?

Practice 4

Two sides of a triangle measure $8$ and $13$ . How many whole-number lengths are possible for the third side?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

Two sides of a triangle are $4$ and $9$ . What is the largest whole-number length the third side can have?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

An isosceles triangle has sides of length $5$ and $11$ (and a third side equal to one of these). What is its perimeter?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

In $\triangle ABC$ , $m\angle A = (3x)^\circ$ , $m\angle B = (2x + 10)^\circ$ and $m\angle C = (x + 20)^\circ$ . Which side is the longest?

Practice 8

In $\triangle JKL$ and $\triangle XYZ$ , $JK = XY = 7$ and $KL = YZ = 10$ . If $m\angle K = 62^\circ$ and $m\angle Y = 85^\circ$ , which statement is true?