Triangle Inequalities

📐 Triangle Inequalities

Part 1 of 5 — The Triangle Inequality Theorem

Topics in This Part

🔑 Key Concept: Not every set of three lengths can close up into a triangle. If two sides are too short, they simply can't reach across the third. The Triangle Inequality Theorem tells you exactly when three lengths will form a triangle.

Can These Sides Make a Triangle?

Imagine two sticks hinged at one end. To form a triangle, their free ends must meet the third side. If the two sticks are too short, they fall short of reaching — no triangle.

• Sides $3$, $4$, $5$ → the two shorter sticks ($3 + 4 = 7$) easily reach across the side of length $5$. ✓ Triangle
• Sides $1$, $2$, $10$ → the two shorter sticks ($1 + 2 = 3$) can't span a gap of $10$. ✗ No triangle

The deciding factor is always whether the two shorter sides together are long enough to stretch across the longest side.

💡 Picture it: if you flatten the triangle completely, the two short sides lie along the long side. To make a real (non-flat) triangle, they must add up to strictly more than the long side.

The Triangle Inequality Theorem

🔑 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 $c$, all three of these must hold:

$a + b > c \qquad a + c > b \qquad b + c > a$

Example: Sides $7$, $10$, $5$

All three pass, so $7$, $10$, $5$ form a triangle.

⚠️ The inequality is strict ($>$, not $\geq$). If a sum equals the third side, the three points lie on a straight line — a degenerate triangle with zero area, which we do not count as a real triangle.

Concept Check 🎯

The One-Check Shortcut

Checking all three inequalities works, but there's a faster way.

🔑 Shortcut: Add the two shortest sides. If that sum is greater than the longest side, the triangle is valid. You only need this one check.

Why? If the two shortest sides already beat the longest side, then any two sides (which can only be longer) will certainly beat the remaining one.

Example: Sides $6$, $9$, $4$

The two shortest are $4$ and $6$. The longest is $9$.

$4 + 6 = 10 > 9 \;\Rightarrow\; \text{valid triangle} ✓$

Example: Sides $3$, $11$, $7$

Two shortest: $3$ and $7$. Longest: $11$.

$3 + 7 = 10 \not> 11 \;\Rightarrow\; \text{NOT a triangle} ✗$

Valid or Not? 🔽

Use the shortcut — add the two shortest sides and compare with the longest.

Apply the Shortcut 🧮

For each set of sides, add the two shortest sides, then decide.

1) Sides $7$, $9$, $4$ — what is the sum of the two shortest sides? 2) Sides $3$, $5$, $9$ — what is the sum of the two shortest sides?

📐 Triangle Inequalities

Part 2 of 5 — The Range of the Third Side

🔑 The Question: If two sides of a triangle are $a$ and $b$, how long can the third side $x$ be? Not anything — it's trapped between a smallest and largest value.

Boxing In the Third Side

Suppose two sides are $a$ and $b$. Let the third side be $x$. The triangle inequality gives us two useful bounds:

• From $a + b > x$: the third side must be less than the sum. So $\;x < a + b$.
• From $a + x > b$ and $b + x > a$: the third side must be more than the difference. So $\;x > |a - b|$.

Putting them together:

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

🔑 Range Rule: The third side is bigger than the difference of the other two and smaller than their sum.

Example: Two sides are $7$ and $10$

$|10 - 7| < x < 10 + 7$ $3 < x < 17$

So the third side can be anything strictly between $3$ and $17$.

Concept Check 🎯

Worked Example: Counting Whole-Number Possibilities

Two sides of a triangle are $4$ and $9$. How many whole-number values are possible for the third side?

Step 1 — Find the range. $|9 - 4| < x < 9 + 4 \;\Rightarrow\; 5 < x < 13$

Step 2 — List the whole numbers strictly between $5$ and $13$.

$x = 6,\; 7,\; 8,\; 9,\; 10,\; 11,\; 12$

Step 3 — Count them. That's $\mathbf{7}$ possible whole-number lengths.

⚠️ The endpoints $5$ and $13$ are excluded — the inequality is strict, so $x = 5$ and $x = 13$ would give degenerate (flat) triangles.

💡 Shortcut for counting whole numbers strictly between two integers $m$ and $n$ (with $m < n$): there are $n - m - 1$ of them. Here $13 - 5 - 1 = 7$. ✓

Find the Range 🧮

For each pair of sides, enter the smallest and largest strict bounds for the third side $x$, where $\text{low} < x < \text{high}$.

1) Sides $9$ and $14$: low $= \,?$ , high $= \,?$ 2) Sides $11$ and $11$: low $= \,?$ , high $= \,?$

Count the Possibilities 🧮

1) Two sides are $5$ and $8$. How many whole-number values are possible for the third side? 2) Two sides are $10$ and $10$. How many whole-number values are possible for the third side?

📐 Triangle Inequalities

Part 3 of 5 — Sides, Angles & Their Order

🔑 The Big Idea: Inside any triangle, the longest side and the largest angle are partners — they always sit opposite each other. The shortest side faces the smallest angle.

The Side–Angle Relationship

🔑 Theorem (Side–Angle Inequality): In any triangle, the larger angle is opposite the longer side, and the longer side is opposite the larger angle. The ordering of the sides exactly matches the ordering of the angles opposite them.

Think of an angle as how "wide open" a corner is. A wider corner forces the side across from it to stretch farther.

Example

In a triangle, suppose the angles are $50°$, $60°$, and $70°$. Then the sides opposite them, from shortest to longest, are:

💡 "Big angle → big side across from it." Just match the order: smallest angle to shortest side, largest angle to longest side.

Reading a Labeled Triangle

A side is named by the two vertices at its ends; the angle "opposite" a side is the vertex not on that side.

In triangle $ABC$:

• Side $BC$ is opposite vertex $A$ (angle $A$).
• Side $AC$ is opposite vertex $B$ (angle $B$).
• Side $AB$ is opposite vertex $C$ (angle $C$).

Example

In triangle $ABC$, $\angle A = 80°$, $\angle B = 60°$, $\angle C = 40°$.

The largest angle is $\angle A$, so the longest side is the one opposite $A$, namely $BC$.

The smallest angle is $\angle C$, so the shortest side is opposite $C$, namely $AB$.

⚠️ A common slip is naming the side touching the big angle. The longest side is the one across from (opposite) the largest angle, not next to it.

Match Side to Angle 🔽

In triangle $ABC$, $\angle A = 75°$, $\angle B = 65°$, $\angle C = 40°$.

Going the Other Way: Sides → Angles

The relationship works in both directions. If you know the sides, you can rank the angles.

Example: Sides $AB = 9$, $BC = 5$, $CA = 7$

Rank the sides shortest → longest: $BC = 5 < CA = 7 < AB = 9$.

The angle opposite each side follows the same order:

• Opposite $BC$ is $\angle A$ → smallest angle.
• Opposite $CA$ is $\angle B$ → middle angle.
• Opposite $AB$ is $\angle C$ → largest angle.

So $\angle A < \angle B < \angle C$.

💡 To rank angles from side lengths: find which side each angle is opposite, then the angles line up in the same order as those opposite sides.

Concept Check 🎯

Largest and Smallest 🧮

A triangle has angles $55°$, $48°$, and $77°$ (in degrees).

1) The longest side is opposite the angle measuring how many degrees? 2) The shortest side is opposite the angle measuring how many degrees?

📐 Triangle Inequalities

Part 4 of 5 — Exterior Angles & Ordering in Practice

🔑 Two more tools: the Exterior Angle Inequality (an exterior angle beats each far interior angle) and a reliable routine for ordering sides when you only know the angles — even when one angle is hidden.

The Exterior Angle Inequality

Extend one side of a triangle past a vertex; the angle formed outside is an exterior angle. The two interior angles not next to it are its remote interior angles.

🔑 Exterior Angle Inequality: An exterior angle of a triangle is greater than either of its two remote interior angles.

This is actually a consequence of the Exterior Angle Theorem, which says the exterior angle equals the sum of the two remote interior angles:

$\text{exterior} = \text{remote}_1 + \text{remote}_2$

Since each remote angle is positive, the exterior angle must exceed each one individually.

Example

A triangle has remote interior angles of $45°$ and $70°$ for a given exterior angle.

$\text{exterior} = 45° + 70° = 115°$

And indeed $115° > 45°$ and $115° > 70°$. ✓

💡 The exterior angle and the interior angle at the same vertex are supplementary (they add to $180°$). Here the interior angle is $180° - 115° = 65°$.

Exterior Angles 🧮

A triangle has interior angles $A$, $B$, and $C$.

1) The remote interior angles for the exterior angle at $C$ are $A = 55°$ and $B = 80°$. What is that exterior angle (in degrees)? 2) Using your answer, what is interior angle $C$ (in degrees)?

Ordering Sides When an Angle Is Missing

Often a problem gives two angles and asks you to order the sides. The trick: the three angles of a triangle always sum to $180°$, so the missing angle is $180°$ minus the other two.

Example: Angles $40°$ and $75°$ are given

Find the third angle:

$180° - 40° - 75° = 65°$

Now order the angles: $40° < 65° < 75°$.

The sides opposite, from shortest to longest, follow that exact order:

• shortest side opposite $40°$,
• middle side opposite $65°$,
• longest side opposite $75°$.

⚠️ Don't forget to find the missing angle first. Comparing only the two given angles can hide which side is truly longest.

Order Everything 🔽

In triangle $XYZ$, $\angle X = 35°$ and $\angle Y = 95°$.

Concept Check 🎯

📐 Triangle Inequalities

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) test whether three lengths form a triangle, (2) find the range for a missing side, (3) match sides to angles, and (4) use exterior angles and ordering. Let's put it all together.

Quick Reference

⚠️ Two habits that prevent most mistakes: use strict inequalities ($>$, never $\geq$), and always pair a side with the angle opposite it — never the angle beside it.

Mixed Practice 🎯

One More Range 🧮

Two sides of a triangle measure $13$ and $6$.

1) Smallest strict bound for the third side $x$ (the low end): $\,?$ 2) Largest strict bound for the third side $x$ (the high end): $\,?$ 3) Number of whole-number values possible for $x$: $\,?$

Exit Quiz ✅

Answer all three to finish the lesson.