Triangle Inequality and Side Angle Relationships Cheat Sheet

This cheat sheet covers the rules that connect the side lengths and angle measures of triangles. Students need these rules to decide whether three lengths can form a triangle and to compare sides or angles without drawing perfectly accurate diagrams. These ideas are important in geometry proofs, construction problems, and coordinate geometry.

They also help students check whether an answer is reasonable before doing more work.

The triangle inequality says each pair of sides in a triangle must add to more than the third side. Side-angle relationships say the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle. The exterior angle theorem connects an outside angle to the two nonadjacent interior angles.

Together, these rules let students rank sides and angles, find possible ranges, and identify impossible triangles.

Key Facts

Three side lengths $a$ , $b$ , and $c$ form a triangle only if $a + b > c$ , $a + c > b$ , and $b + c > a$ .
When side lengths are ordered as $a < b < c$ , the opposite angles are ordered as $A < B < C$ .
When angle measures are ordered as $A < B < C$ , the opposite side lengths are ordered as $a < b < c$ .
The largest angle of a triangle is always opposite the longest side.
The smallest angle of a triangle is always opposite the shortest side.
The sum of the interior angles of any triangle is $180^\circ$ .
An exterior angle of a triangle equals the sum of the two remote interior angles, so $m\angle 1 = m\angle 2 + m\angle 3$ .
If two sides of a triangle are $a$ and $b$ , then the third side $x$ must satisfy $\left|a - b\right| < x < a + b$ .

Vocabulary

Triangle inequality: The rule that the sum of any two side lengths of a triangle must be greater than the third side length.
Opposite side: The side across from a given angle in a triangle.
Opposite angle: The angle across from a given side in a triangle.
Exterior angle: An angle formed by one side of a triangle and the extension of an adjacent side.
Remote interior angles: The two interior angles of a triangle that are not adjacent to a given exterior angle.
Included side: The side located between two specified angles of a triangle.

Common Mistakes to Avoid

Checking only one triangle inequality, which is wrong because all three inequalities $a + b > c$ , $a + c > b$ , and $b + c > a$ must be true.
Using $a + b \ge c$ instead of $a + b > c$ , which is wrong because equality makes a straight line, not a triangle.
Matching the largest angle with the shortest side, which is wrong because the largest angle is always opposite the longest side.
Assuming a diagram is drawn to scale, which is wrong because geometry diagrams may exaggerate or shrink sides and angles.
Forgetting the absolute value in the third-side range, which is wrong because the lower bound must be the positive difference $\left|a - b\right|$ .

Practice Questions

1 Can side lengths $5$ , $7$ , and $13$ form a triangle? Explain using the triangle inequality.
2 Two sides of a triangle are $8$ cm and $11$ cm. Write the possible range for the third side $x$ .
3 In $\triangle ABC$ , $m\angle A = 45^\circ$ , $m\angle B = 60^\circ$ , and $m\angle C = 75^\circ$ . Rank the side lengths $AB$ , $BC$ , and $AC$ from shortest to longest.
4 A triangle has one side much longer than the other two. Explain what must be true about the angle opposite that longest side.

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Triangle Inequality and Side Angle Relationships Cheat Sheet

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